UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


ELEMENTS  OF  GEODESY. 


BY 

J.   HOWARD   GORE,   B.S.,  PH.D., 

Professor  of  Mathematics  in  The  Columbian   University;    sometime  Astronomer 

and  Topographer  U.  S.  Geological  Survey;  Acting  Assistant  U,  S. 

Coast  and  Geodetic  Survey;  Associate  des  Preussischen 

Geoadtischen   Institutes. 


SECOND  EDITION.     REVISED  AND  CORRECTED. 


NEW  YORK: 

JOHN     WILEY     &     SONS, 

15   ASTOR    PLACE. 

1889. 


COPYRIGHT,  1886, 
BY   IOHN   WILEY   &   SONS 


3>o 


PREFACE. 


THE  chief  reason  for  making  the  following  pages  public  is 
the  desire  to  put  into  better  shape  the  principles  of  Geodesy, 
and  have  accessible  in  a  single  book  what  heretofore  has  been 
scattered  through   many.     The  advanced   student  and   prac- 
tised observer  will  find  nothing  new  in  this  work,  and  may, 
when  accident  throws  it  into  their  hands,  lay  it  aside  with  feel- 
»    ings  of   disappointment.     But  it  is  hoped  that  the  beginner 
,    will  be  enabled  to  get  a  clear  insight  into  the  subject,  and  feel 
*    grateful  that  the  discoveries  and  writings  of  many  have  been 
^   so  condensed  or  elaborated  as  to  make  the  study  of  Geodesy 
pleasant.     The  plan  pursued  in  the  discussions  that  follow  is 
Pto  take  up  each  division  in  its  logical  order,  develop  each  for- 
*J  mula  step  by  step,  and  leave  the*  results  or  conclusion  in  the 
shape  that  the  majority  of  writers  have  considered  the  best. 
In  the  text  only  occasional  acknowledgments  have  been  in- 
serted, though  at  the  end  of  each  chapter  a  list  of  books  will 
be  found  to  which  reference  has  been  frequently  made.     These 
lists  are  by  no  means  complete,  so  far  as  the  literature  of  the 
subject  is  concerned,  but  contain  the  titles  of  those  books 
which  were  found  the  most    helpful  while  engaged   in    self- 
instruction.     The  compilation  of  a  complete  Bibliography  is 


jv  PREFACE. 

now  in  hand,  forming  a  part  of  a  History  of  Geodesy,  which 
will  be  finished  in  the  course  of  a  few  years. 

It  is  a  pleasure  to  record  the  interest  of  Mr.  Henry  Gan- 
nett, Chief  Geographer  of  the  U.  S.  Geological  Survey,  which 
prompted  him  to  read  the  manuscript  and  suggest  important 
improvements. 

I  desire  to  acknowledge  my  obligations  to  my  associate,  Pro- 
fessor H.  L.  Hodgkins,  A.M.,  for  the  interest  he  has  shown  in 
the  work,  and  for  his  careful  revision  of  the  proof-sheets  as 
they  came  from  the  press. 

I  also  wish  to  express  my  indebtedness  to  my  friend  Miss 
Lizzie  P.  Brown  for  her  suggestions,  and  for  the  elimination  of 
errors  that  otherwise  would  have  seriously  blemished  the  work. 
It  is  hoped  that  errors  do  not  remain  in  sufficient  number  or 
of  such  size  as  to  impair  the  clearness  or  accuracy  of  the  dis- 
cussions that  follow. 

When  page  102  was  written,  it  was  thought  that  a  satisfac- 
tory formula  could  be  procured  for  the  computation  referred 
to,  but  the  increasing  doubts  regarding  the  coefficient  of  re- 
fraction have  induced  me  to  omit  further  consideration  of  the 
subject. 

WASHINGTON,  July,  1886. 


GEODETIC    OPERATIONS. 


CHAPTER   I. 

AN   HISTORIC  SKETCH   OF  GEODETIC   OPERATIONS. 

ONE  of  the  first  problems  that  suggested  itself  for  solution 
in  the  intellectual  infancy  of  mankind  was:  "What  is  the 
earth,  its  size  and  shape?"  The  possibility  of  examining  the 
constituency  of  the  superficial  strata  answered  with  sufficient 
exactness,  for  the  time  being,  the  first  part  of  the  question. 
The  natural  conclusion  deducible  from  daily  experience  and 
observation  is :  were  the  earth  deprived  of  the  irregularities 
produced  by  the  valleys  and  mountains,  its  surface  would  be 
a  plane.  The  exact  date  of  the  abandonment  of  this  theory 
is  unknown.  Froriep  refers  to  a  Sanskrit  manuscript  contain- 
ing the  following  sentence:  "According  to  the  Chaldeans, 
4000  steps  of  a  camel  make  a  mile,  66f  miles  a  degree,  from 
which  the  circumference  of  the  earth  is  24,000  miles."  Of  the 
authenticated  announcements  of  hypotheses,  Pythagoras  was 
the  first  to  declare  that  the  earth  is  spherical.  This  honor 
is  sometimes  assigned  to  Thales  and  Anaximander.  Archi- 
medes gave  as  an  approximate  value  for  the  circumference 
300,000  stadia.  To  Eratosthenes  (B.C.  276)  belongs  the  credit 
of  making  the  initial  step  towards  a  determination  of  the  cir- 
cumference. He  observed  that  at  Syene,  in  Southern  Egypt, 
an  object  on  the  day  of  summer  solstice  cast  no  shadow,  while 
1 


2  GEODETIC  OPERATIONS. 

at  Alexandria  the  sun  made  an  angle  with  the  vertical  equal  to 
one  fiftieth  of  a  circumference.  Considering  that  Alexandria 
was  north  of  Syene,  he  reasoned  that  the  entire  circumference 
of  the  earth  was  50  times  the  distance  between  those  places,  or 
250,000  stadia;  this  he  afterwards  increased  to  252,000  stadia. 
The  neglect  of  the  sun's  diameter  in  the  determination  of  dec- 
lination, and  the  false  supposition  that  Alexandria  and  Syene 
were  on  the  same  meridian,  introduced  considerable  inaccura- 
cies in  his  results,  the  exact  amount  of  which,  however,  \ve  can- 
not estimate  owing  to  our  ignorance  as  to  the  length  of  the 
stadium. 

About  two  hundred  years  later  Posidonius  determined  the 
amplitude  of  the  arc  between  Rhodes  and  Alexandria  from 
observations  on  the  star  Canopus  at  both  places.  At  Rhodes 
he  saw  this  star,  when  on  the  meridian,  just  visible  above  the 
horizon,  and  at  Alexandria  its  altitude  at  the  same  time  was 
•fa  of  a  great  circle.  From  this  he  concluded  that  the  circum- 
ference was  48  times  the  distance  these  places  were  apart,  or 
48  X  500O  stadia  =  240,000  stadia.  If  we  know  the  latitude 
of  two  points  on  the  same  meridian,  the  difference  will  be  the 
amplitude  of  the  arc  passing  through  them,  and  the  circumfer- 
ence will  bear  the  same  ratio  to  the  length  of  the  arc  that  its 
amplitude  bears  to  four  right  angles. 

Letronne  has  shown  that  the  amplitude  of  the  arc  Posi- 
donius used  is  only  5°  =  ^  of  a  great  circle,  and  Strabo  gives 
40CO  stadia  as  the  length  of  the  arc,  making  the  circumference 
288,000  stadia. 

Ptolemy  in  the  second  century  gave  180,000  stadia  for  the 
circumference,  but  does  not  state  his  authority.  Posch  infers 
that  it  was  taken  from  the  Chaldean  value,  since  Ptolemy  gives 
a  Chaldean  mile  equal  to  ;£  stadia,  and  ;£  times  24,000  = 
180.000.  In  827  an  Arabian  caliph  imposed  upon  his  astrono- 
mers the  task  of  measuring  an  arc,  and  of  deducing  from  it 
the  length  of  the  circumference  of  the  earth. 


HISTORIC  SKETCH.  3 

Abulfeda  in  1322  gave  the  following  description  of  the 
method  employed  by  them  :  There  were  two  parties  ;  one  start- 
ing from  a  fixed  point  measured  a  line  due  north  with  a  rod, 
the  other  party  going  due  south  ;  both  continuing  until  the  ob- 
served latitudes  were  found  to  differ  by  one  degree  from  that 
of  the  starting-point.  The  first  party  found  56  miles  and  the 
second  56!  miles  for  a  degree.  The  latter  result  was  accepted, 
its  equivalent  being  approximately  71  English  miles. 

This  was  a  great  improvement  upon  the  methods  of  the 
Grecians,  who  estimated  their  distances  by  days'  marches  of  so 
many  stadia  a  day. 

Fernel  in  1525  made  a  measurement  for  the  determination 
of  the  length  of  a  degree  by  counting  the  number  of  revolu- 
tions made  by  a  wheel  of  known  circumference  in  going  from 
Paris  to  Amiens.  He  applied  a  correction  to  reduce  the  broken 
line  to  a  straight  one,  and  the  latitude  observations  were  made 
with  a  5-foot  sector,  giving  for  a  degree  365,088  English  feet. 
A  few  years  later  Father  Riccioli  made  an  arc-determination 
in  Italy,  but  it  was  too  short  to  be  of  any  importance.  The 
first  attempt  to  determine  the  size  of  the  earth  by  means  of 
triangulation  was  by  Willebrord  Snellius  in  1615.  He  measured 
a  base-line  with  a  chain  between  Leyden  and  Soeterwood,  and 
connected  it  by  means  of  triangles,  33  in  number,  so  as  to  com- 
pute the  distance  from  Alcmaar  to  Bergen-op-Zoom.  This 
distance  he  reduced  to  its  equivalent  along  a  meridian,  giving 
an  arc  of  i°  ii'  05"  amplitude,  from  which  he  found  55,074 
toises  for  a  degree  (a  toise  being  equal  to  6.3946  English  feet). 
Kastner  has  shown  that  the  neglect  of  spherical  excess  in  the 
reduction  of  these  triangles  causes  an  error  of  nearly  a  toise. 
In  1722  the  measurements  were  repeated,  using  for  the  angle- 
determinations  a  sector  of  5  feet  radius;  this  second  reduction 
gave  57,033  toises  for  a  degree.  One  can  scarcely  conceive  of 
the  amount  of  labor  such  an  undertaking  necessitated  at  a 
time  when  there  were  no  logarithmic  tables  to  lighten  the  work. 


A  GEODETIC  OPERATIONS. 

Norwood  in  1635  measured  with  a  chain  the  distance  from 
London  to  York,  obtaining  for  a  degree  57,424  toises. 

In  the  measurement  of  angles  Snellius  had  sights  attached 
to  his  sector,  making  a  close  reading  impracticable. 

While  the  telescope  was  made  use  of  as  early  as  1608,  no 
one  had  thought  of  putting  it  on  an  angle-reading  instrument 
until  Picard,  in  1669,  placed  in  the  focus  of  a  telescope  spider- 
lines  to  mark  the  optical  axis,  which,  according  to  some  au- 
thorities, had  already  been  done  by  Gascoigne  in  1640.  He 
measured  a  base-line  nearly  7  miles  long,  and  with  a  sector  of 
10  feet  radius,  to  which  was  attached  a  telescope,  the  angles 
were  carefully  read,  until  Malvoisine  and  Amiens  were  con- 
nected by  a  chain  of  triangles.  This  gave  an  arc  of  1°  22'  58", 
from  which  he  computed  57,060  toises  as  the  length  of  a 
degree.  At  this  time  the  effect  of  aberration  and  nutation 
were  unknown,  which,  if  allowed  for,  would  have  shortened 
his  arc  by  3".  However,  when  his  unit  of  linear  measure  was 
more  accurately  compared  with  the  standard  it  was  found  to 
be  too  short,  so  that  when  Lacaille  revised  the  work  he  ob- 
tained the  identical  result  that  Picard  had  previously  an- 
nounced. 

The  uncertainty  of  ascertaining  the  circumference  of  the 
earth  from  so  short  an  arc  was  so  keenly  felt  at  this  time  that 
the  extension  of  this  arc  both  northward  and  southward  was 
undertaken  by  the  Cassini,  father  and  son,  Lahire,  and  Maraldi, 
carrying  it  from  Paris  to  Dunkirk,  and  from  Paris  to  Perpignan, 
the  entire  arc  being  about  8°  31'. 

The  published  results  of  Picard's  work  were  rendered  famous 
by  endorsing  Newton's  hypothesis  of  universal  gravitation. 
Newton  had  attempted  to  prove  this  theory  by  comparing  the 
force  of  gravity  on  a  body  at  the  moon's  distance  with  the 
power  required  to  keep  her  in  her  orbit.  He  used  in  his  com- 
putations the  diameter  of  the  earth  as  somewhat  less  than 
7000  miles.  The  result  failed  to  show  the  analogy  he  had  con- 


HISTORIC  SKETCH,  5 

ceived  ;  so  he  laid  aside  his  theory,  so  brilliant  in  conception,  so 
lacking  in  verification.  But  twenty  years  later,  when  Picard's 
length  of  a  degree  was  made  known,  increasing  the  diameter  of 
the  earth  by  about  a  thousand  miles,  Newton  was  able  to  show 
that  the  deflection  of  the  orbit  of  the  moon  from  a  straight 
line  was  equivalent  to  a  fall  of  16  feet  in  one  minute,  the  same 
distance  through  which  a  body  falls  in  one  second  at  the  sur- 
face of  the  earth.  The  distance  fallen  being  as  the  square  of 
the  time,  it  followed  that  the  force  of  gravity  at  the  surface  of 
the  earth  is  3600  times  as  great  as  the  force  which  holds  the 
moon  in  her  orbit.  This  number  is  the  square  of  60,  which 
therefore  expresses  the  number  of  times  the  moon  is  more  dis- 
tant from  the  centre  of  the  earth  than  we  are.  If  with  the 
rude  means  employed  by  Picard  his  errors  had  not  eliminated 
one  another,  or  if  their  extent  had  been  discovered  without 
knowing  their  compensating  character,  the  undemonstrated 
law  of  gravitation  would  have  remained  as  an  hypothesis,  ce- 
lestial mechanics  would  have  been  without  the  mainspring  of 
its  existence,  and  we  would  now  be  groping  in  the  darkness  of 
an  antecedent  century. 

Newton  also  maintained  that,  owing  to  the  greater  centrifugal 
force  of  the  particles  at  the  equator,  a  meridian  section  of  the 
earth  would  be  an  oblate  ellipse  ;  that  is,  the  equatorial  axis 
would  exceed  the  polar.  If  such  were  the  case,  the  radius  of 
curvature  would  increase  in  going  from  the  equator  towards 
the  pole ;  and  as  the  co-latitude  is  the  angle  formed  by  the 
normal  with  the  polar  axis,  if  the  normal  increases,  the  arc  of  a 
constant  angle  must  become  larger,  therefore  the  oblate  hy- 
pothesis requires  for  verification  that  the  degrees  increase  in 
going  from  the  equator  towards  either  pole.  Consequently  the 
results  of  Cassini's  long  arc  determination  were  awaited  with 
impatience,  until  1718,  when  the  announcement  was  made 
that  the  northern  degree  was  shorter  than  the  southern ; 
this  pleased  the  French,  as  it  gave  them  an  opportunity  to 


6  GEODETIC  OPERATIONS. 

again  say  that  the  country  across  the  Channel  was  a  "  Naza- 
reth from  which  no  good  thing  could  come."  A  degree  of 
the  northern  arc  gave  56,960  toises,  and  of  the  southern 
57,098  toises,  from  which  it  appeared  that  the  earth  was  pro- 
late. 

Huygens  in  1691  published  his  theory  regarding  centrifu- 
gal motion,  describing  experiments  that  proved  that  a  rotat- 
ing mass  like  the  earth  would  have  its  greater  axis  perpendicu- 
lar to  the  axis  of  rotation.  Hence  the  terrestrial  degrees 
increase  northward.  It  was  a  part  of  Newton's  theory  that  as 
the  polar  diameter  is  less  than  the  equatorial,  the  force  of 
gravity  must  increase  in  going  towards  the  pole,  and  therefore 
a  clock  regulated  by  a  pendulum  would  lose  time  when  carried 
towards  the  equator.  When  Richer  returned  in  1672  from  the 
Island  of  Cayenne,  where  he  had  been  sent  to  make  astronomic 
observations,  he  found  that  his  clock  while  at  the  island  lost 
two  minutes  a  day  when  compared  with  its  rate  at  Paris,  and, 
furthermore,  the  length  of  his  pendulum  beating  seconds  was 
l^  lines  shorter  than  the  Paris  seconds  pendulum,  showing  that 
Cayenne  was  farther  than  Paris  from  the  centre  of  the  earth. 
A  portion  of  this  difference  in  the  lengths  of  the  pendulums 
was  supposed  to  be  due  to  increased  counteracting  effect  of 
centrifugal  force  nearer  the  equator,  but  Newton  showed  that 
the  discrepancy  was  too  great  for  a  spherical  globe.  Varin 
and  Des  Hays  had  a  similar  experience  with  pendulums  taken 
to  points  almost  under  the  equator. 

Under  the  excitement  occasioned  by  this  sharp  controversy,  as 
well  as  from  a  desire  to  know  the  truth,  the  French  Academy 
decided  to  submit  the  problem  to  a  most  crucial  test  by  meas- 
uring one  arc  crossing  the  equator,  and  another  within  the 
polar  circle.  Knowing  the  fierce  criticism  that  would  be 
brought  to  bear  upon  every  feature  of  the  work,  the  partici- 
pants determined  to  use  the  most  refined  instruments  and 
most  approved  methods.  In  May,  1735,  an  expedition  consist- 


HISTORIC  SKETCH.  7 

ing  of  Godin,  Bouguer,  De  la  Condamine,  and  Ulloa  set  out  for 
Peru.  The  base  was  selected  near  Quito  at  an  elevation  of 
nearly  8000  feet  above  sea-level.  Its  length  was  7.6  miles  as 
deduced  from  a  duplicate  measurement,  made  by  two  parties 
working  in  opposite  directions.  The  measuring-rods  were  of 
wood,  twenty  feet  in  length,  terminated  at  either  end  in  copper 
tips  to  prevent  wearing  by  attrition. 

They  were  laid  approximately  horizontal,  the  deviation 
therefrom  being  estimated  by  a  plummet  swinging  over  a 
graduated  arc.  A  comparison  with  a  field  standard  was  made 
each  day,  this  standard  being  laid  off  from  the  toise  taken  from 
Paris,  which  afterwards  became  the  legal  unit  in  France,  and  is 
known  as  the  Toise  of  Peru.  The  angles  of  the  33  triangles 
were  measured  on  quadrants  of  2  and  3  feet  radius ;  these 
were  so  defective,  however,  that  great  care  was  necessary  in  de- 
termining the  instrumental  errors  and  applying  them  to  each 
angle-determination.  Twenty  observations  were  made  at  dif- 
ferent stations  for  ascertaining  the  azimuths. 

The  amplitude  of  the  arc  was  found  from  simultaneous  lati- 
tude-observations made  at  the  terminal  stations  on  the  same 
•star.  Realizing  that  great  uncertainties  would  arise  from  a 
faulty  determination  of  the  amplitude,  the  latitude-observa- 
tions were  made  with  sectors  12  and  8  feet  radius,  on  the  sup- 
position that  the  larger  the  sector  the  more  accurate  would  be 
the  results.  But  the  instability  of  the  supports  allowed  such 
great  flexure  that  they  were  almost  wholly  reconstructed  on 
the  field. 

A  southern  base  was  measured  as  a  check  near  Cotopaxi 
at  an  elevation  of  nearly  10,000  feet  above  sea-level.  Its 
length,  6.4  miles,  as  measured,  differed  from  the  value  com- 
puted from  the  northern  base  by  only  one  toise,  and  the  entire 
arc  was  but  ten  toises  longer  according  to  Condamine  than 
found  by  Bouguer.  The  amplitude  as  deduced  by  Bouguer 
was  3°  7'  i",  giving  for  the  length  of  a  degree  reduced  to  sea 


8  GEODETIC  OPERATIONS. 

level  56,753  toises — the  mean  of  the  two  computations  just 
quoted.  The  field-work  occupied  two  years,  but  the  results 
were  not  published  until  the  beginning  of  1746. 

Von  Zach  revised  the  calculations,  finding  the  arc  to  be  71 
toises  shorter;  and  Delambre  recomputed  the  latitudes,  from 
which  he  found  the  amplitude  increased  by  a  little  more  than 
2  seconds.  According  to  the  former,  a  degree  would  have  at 
that  latitude  a  length  of  56,731  toises,  while  the  latter  would 
give  56,737  toises,  a  value  indorsed  by  Arago. 

The  polar  party,  consisting  of  Maupertuis,  Clairault,  Camus, 
Le  Monnier,  Outhier,  and  Celsius,  Professor  of  Astronomy  at 
Upsal,  reached  its  destinaton  May  21,  1736.  The  river  Tornea, 
flowing  south,  with  mountains  of  greater  or  less  elevation  on 
each  side,  afforded  in  its  valley  a  suitable  location  for  the  base, 
and  the  mountains,  points  for  the  triangle  stations.  The 
signals  were  built  of  trees  stripped  of  their  bark,  in  the  shape 
of  a  hollow  cone.  The  angles  were  measured  with  a  quadrant 
of  2  feet  radius  provided  with  a  micrometer,  each  angle  being 
read  by  more  than  one  person,  the  average  of  the  means  of  the 
individual  results  being  taken.  Great  care  was  exercised  in  cen- 
tring the  instrument  and  in  checking  the  readings  by  observ- 
ing additional  angles  whose  sums  or  differences  would  give  the 
angles  wanted. 

Latitude  observations  were  made  by  determining  the  differ- 
ence of  zenith  distances  of  two  stars  with  a  sector  consisting  of 
a  telescope  9  feet  long,  which  formed  the  radius  of  an  arc  5°  30'. 
This  arc  was  divided  into  spaces  of  f  30",  which  were  subdi- 
vided by  a  micrometer.  From  the  observations  corrected  for 
aberration,  nutation,  and  precession,  the  amplitude  was  found 
to  be  57'  26^.93  according  to  Outhier,  57'  28*75  according  to 
Maupertuis,  and  57'  28*.5  as  given  by  Celsius.  The  base  was 
measured  during  the  winter  over  the  frozen  snow  and  ice  on 
the  river  Tornea,  the  terminal  points  only  being  on  land.  The 
measuring-bars  were  of  wood,  each  30  feet  long,  as  determined 


HISTORIC  SKETCH.  9 

by  comparison  with  an  iron  toise  carried  from  Paris.  Daily 
comparisons  were  made  by  placing  the  rods  between  two  iron 
nails,  previously  driven  at  a  distance  apart  just  equal  to  the 
length  of  one  of  the  rods  on  the  first  day.  It  was  found  that 
they  had  not  changed  in  length  during  the  work. 

There  were  two  parties,  each  having  four  rods,  which  they 
placed  end  to  end  on  the  snow.  In  this  manner  the  entire 
base  was  measured  twice,  both  parties  laying  the  same  number 
of  bars  each  day  giving  a  daily  check.  The  total  difference  in 
the  two  results  was  only  4  inches  in  a  distance  of  8.9  miles,  a 
degree  of  accuracy  that  is  quite  remarkable  when  it  is  consid- 
ered that  the  average  temperature  was  6  degrees  F.  below 
zero.  From  this  arc  a  degree  cut  by  the  polar  circle  was  ascer- 
tained to  be  57,437  toises.  While  many  precautions  were  taken, 
the  disagreement  in  the  astronomic  reductions,  and  some  in- 
strumental errors  that  were  afterwards  discovered,  caused  some 
doubt  as  to  the  reliability  of  the  work.  If  correct,  a  degree  at 
this  point  would  be  377  toises  longer  than  a  degree  at  Paris,  a 
difference  greater  than  the  theorists  had  calculated,  and  more 
confirmatory  of  the  oblate  hypothesis  than  was  wanted. 

Cassini,  De  Thuri,  and  Lacaille  revised  the  French  arc  pre- 
viously measured  by  J.  and  D.  Cassini,  and,  comparing  the 
northern  with  the  southern  portion  of  the  arc,  they  declared 
that  the  earth  was  oblate;  this  was  announced  in  1744.  In 
1743,  Clairaut,  reasoning  that  the  earth,  instead  of  being  of  uni- 
form density,  each  particle  being  pressed  down  by  all  that  is 
above,  those  near  the  centre  must  be  denser  than  those  nearer 
the  surface.  Starting  with  the  hypothesis  that  the  density  is 
a  function  of  the  distance  from  the  surface,  he  declared  that 
the  earth  was  oblate,  but  not  to  the  extent  that  Newton  had 
supposed. 

Let  us,  in  review,  contemplate  the  condition  of  this  problem 
at  this  period  :  Newton,  in  1687,  from  a  theoretic  analysis,  said 
the  earth  was  oblate ;  this  explained  the  behavior  of  Richer's 


IO  GEODETIC  OPERATIONS. 

clock  in  1672.  Huygens,  in  1691,  revolved  a  hollow  metallic  globe, 
and  saw  it  protrude  at  the  centre ;  hence,  from  analogy,  he  ac- 
cepted the  oblate  hypothesis.  Cassini's  arc  of  1718  declared 
the  theorists  wrong.  The  Lapland  labors  of  Maupertuis,  nine- 
teen years  later,  negatived  Cassini's  conclusion.  Clairaut,  in 
1743,  endorsed  Maupertuis,  but  failed  to  show  so  great  an  ob- 
lateness.  In  1744,  Lacaille,  repeating  the  work  of  Cassini, 
changed  the  results  until  they  conformed  to  theory ;  and  hardly 
a  year  later  came  the  fruit  of  the  ten  years'  labor  in  Peru  to  as- 
sert that  Newton,  Huygens,  and  Clairaut  were  all  right,  in  dif- 
ferent degrees. 

Lacaille,  in  1750,  went  to  the  Cape  of  Good  Hope  to  deter- 
mine the  moon's  parallax,  and  while  there  he  measured  an  arc 
of  [^degrees  in  south  latitude  33°  iS£',  from  which  he  deduced 
57,037  toises  as  the  length  of  a  degree.  The  short  time  de- 
voted to  this  work,  and  the  inferior  quality  of  his  instruments, 
caused  this  determination  to  be  lightly  regarded.  The  next 
triangulation  was  executed  by  Boscovich  in  1751-53,  in  latitude 
43°  N.,  where  an  arc  of  2°  gave  56,973  toises  as  the  length  of 
a  degree.  In  1768  Beccaria  found  57,024  toises  for  a  degree 
in  latitude  44°  44'  N.  Zach  revised  this  work  and  found  a  dif- 
ference of  15  toises  in  the  length  of  the  arc,  and  numerous 
errors  in  the  angle-measurements.  Also  the  proximity  of  the 
northern  terminus  of  the  arc  to  the  mountains  suggests  that 
the  unnoticed  deflection  of  the  plumb-line  gave  to  the  arc  a 
wrong  amplitude. 

In  connection  with  Liesganig,  the  indefatigable  Boscovich 
measured  an  arc  of  3°,  giving  for  the  northern  portion  in  lati- 
tude 48°  43',  57,086  toises  for  a  degree,  and  for  the  southern 
part  they  found  a  degree  to  be  56,881  toises— a  difference  too 
great  to  give  to  the  work  much  confidence. 

The  surveyors  Mason  and  Dixon  (1764-68),  in  locating  the 
boundary-line  between  the  properties  of  the  Penn  family  and 
Lord  Baltimore,  a  portion  of  which  afterwards  became  the 


HISTORIC  SKETCH.  II 

boundary-line  between  Pennsylvania  and  Maryland,  saw  that 
that  part  of  the  line  separating  Maryland  from  Delaware  was 
located  on  low  and  level  land,  almost  coinciding  with  a  merid- 
ian. For  this  reason  they  concluded  that  it  would  be  suitable 
for  measuring  the  length  of  a  degree.  The  Royal  Society  of 
London  voted  them  money  for  the  work.  The  whole  distance 
was  measured  with  wooden  rods  20  feet  in  length ;  contact  was 
carefully  made  with  rods  level,  and  thermometric  readings 
made  to  correct  for  expansion.  Latitude  was  ascertained 
from  equal  zenith-distance  observations,  and  azimuth  meas- 
ured from  a  meridian  mark  determined  from  astronomic  obser- 
vations. 

The  amplitude  of  the  arc  was  i°  28' 45",  and  the  length  as 
measured  gave  for  a  degree  56,888  toises. 

In  1783  the  proposition  was  made  on  the  part  of  the  French 
geodesists  to  unite  Paris  and  Greenwich  by,  triangulation. 
General  Roy  was  placed  in  charge  of  the  operations  on  the 
English  side  of  the  Channel,  and  Count  Cassini,  Mechain,  and 
Legendre  attended  to  that  part  of  the  work  that  fell  within 
France.  In  this  work  every  precaution  was  taken  to  secure 
good  results,  and  all  refinements  at  that  time  devised  were 
utilized.  For  the  first  time  Ramsden's  theodolite  with  a  circle 
of  3  feet  in  diameter  was  employed  in  measuring  the  angles. 

This  circle  was  divided  into  15-minute  spaces,  and  was  read 
at  three  points  by  micrometers  rigidly  connected  with  one 
another.  The  telescope  had  a  focal  length  of  three  feet,  and  of 
sufficient  power  to  render  visible  a  church-tower  at  a  distance 
of  forty-eight  miles  across  water.  The  history  of  this  the- 
odolite would  form  a  large  part  of  the  history  of  the  English 
triangulation.  Sir  Henry  James,  in  speaking  of  it  in  1863, 
said  :  "  When  it  is  considered  that  this  instrument  has  been 
in  use  for  the  last  seventy-five  years,  and  that  it  has  been 
placed  upon  many  of  our  very  highest  mountains,  on  our  most 
distant  islands,  and  on  the  pinnacles  of  our  loftiest  churches,  the 


12  GEODETIC  OPERATIONS. 

perfection  with  which  this  instrument  was  made,  and  the  care 
with  which  it  has  been  preserved,  is  truly  remarkable."  Also 
Colonel  Clarke,  in  1880,  remarks  that  it  is  as  good  as  when  it 
left  the  workshop. 

The  triangulation  in  England  rested. upon  the  Hounslow 
Heath  base.  The  first  measurement  of  this  base  was  made  in 
June,  1784,  with  a  steel  chain  of  100  feet  in  length,  giving  for 
the  length  of  the  line,  corrected  for  temperature,  27,408.22  feet. 
A  second  determination  was  made  using  wooden  rods,  termi- 
nating in  bell-metal  tips,  the  entire  length  being  20  feet  3  inches. 
In  the  course  of  the  work  it  was  noticed  that  the  rods  were 
affected  by  moisture  so  as  to  render  the  results,  27,406.26  feet, 
unreliable.  At  the  suggestion  of  Colonel  Calderwood,  it  was 
decided  to  measure  the  line  with  glass  tubes.  These  were  20 
feet  long,  supported  in  wooden  cases  8  inches  deep,  and  con- 
tact was  made  as  in  the  slide-contact  forms.  In  the  reduction 
of  the  length  of  the  base  a  carefully  determined  coefficient  of 
expansion,  .0000043,  was  employed,  giving  for  the  length  of  the 
base  27,404.0137  feet. 

Another  measurement  made  with  a  steel  chain,  using  five 
thermometers  for  temperature-indications,  gave  a  result  differ- 
ing from  the  last  by  only  2  inches.  This  length  was  the  equiva- 
lent reduced  to  sea-level — a  correction  being  applied  for  the 
first  time  in  the  history  of  geodesy. 

In  the  French  work  nothing  new  was  introduced  except  the 
repeating-circle.  This  was  constructed  on  a  principle  pointed 
out  by  Tobias  Mayer,  Professor  in  the  University  of  Gottin- 
gen,  which  was  thought  to  eliminate  errors  of  graduation  that 
had  at  that  time  become  a  source  of  fear,  owing  to  the  imper- 
fect means  for  graduating.  By  the  method  of  repetition  it 
was  supposed  that  if  a  number  of  pointings  be  made  with 
equal  care,  and  the  final  reading  be  divided  by  the  number  of 
pointings,  the  error  of  graduation  as  affecting  the  angle  so  re- 
peated  would  be  likewise  divided,  and  hence  be  too  small  to 


HISTORIC  SKETCH.  13 

be  appreciable.  If  all  the  parts  of  the  instrument  were  rigid, 
and  if  the  circle  or  telescope  could  be  clamped  in  place  without 
the  one  in  its  motion  moving  the  other,  the  theory  might  be 
endorsed  in  practice.  However,  these  conditions  have  never 
been  definitely  secured,  nor  is  it  likely  that  a  clamp  can  be  de- 
vised that  will  not  give  in  its  working  a  travelling  motion. 
These  obstacles  did  not  present  themselves  with  sufficient 
force  to  cause  the  French  to  abandon  this  form  of  angle-read- 
ing instruments  until  it  had  mutilated  their  labors  covering 
a  half-century. 

Barrow,  in  1790,  measured  an  arc  of  i°  8'  in  East  Indies,  ob- 
taining for  a  degree  in  latitude  23°  18",  56,725  toises. 

The  year  1791  carries  with  it  the  honor  of  having  witnessed 
the  inception  of  the  most  majestic  scheme  ever  devised  for  ob- 
taining and  fixing  a  standard  unit  of  measure.  Laplace  and 
Lagrange,  with  the  support  of  the  principal  mathematicians  of 
that  period  in  France,  proposed  to  the  Assembly  of  France 
that  the  standard  linear  unit  should  be  a  ten-millionth  part  of 
the  earth's  quadrant,  to  be  called  a  metre ;  the  length  of  this 
quadrant  to  be  determined  by  the  measurement  of  an  arc  of 
9°  40'  24*,  of  which  nearly  two  thirds  was  north  of  the  45th 
parallel, — the  northern  terminus  being  Dunkirk,  and  the  south- 
ern, Barcelona.  Delambre  was  in  charge  of  the  work  from 
Dunkirk  to  Rodez,  and  Mechain  completed  that  portion  ex- 
tending from  Rodez  to  Barcelona. 

Two  base-lines  were  measured,  one  at  Melun,  near  Paris,  and 
the  other  at  Perpignan,  each  about  seven  and  a  quarter  miles 
long.  The  measuring-bars  were  four  in  number,  each  com- 
posed of  two  strips  of  metal  two  toises  in  length,  half  an  inch 
in  width,  and  a  twelfth  of  an  inch  in  thickness.  The  two 
metal  strips  were  supported  on  a  stout  beam  of  wood,  the 
whole  resting  on  iron  tripods  provided  with  levelling-screws. 

One  of  the  strips  was  made  of  platinum  ;  the  other,  resting  on 
this,  was  copper,  shorter  than  the  platinum  by  about  6  inches. 


!4  GEODETIC  OPERATIONS. 

At  one  end  they  were  firmly  fastened  together,  but  free  to 
move  throughout  the  remainder  of  their  lengths ;  so  that  by 
means  of  a  graduated  scale  on  the  free  end  of  the  copper  and 
a  vernier  on  the  corresponding  end  of  the  platinum,  the  vary- 
ing lengths  owing  to  the  different  expansions  of  the  two 
metals  could  be  determined,  and  hence  the  temperature  known. 
This  was  the  invention  of  Borda,  and  is  now  known  as  the 
Borda  scale,  or  metallic  thermometer.  The  bars  were  compared 
indirectly  with  the  toise  of  Peru  by  their  maker,  and  No.  I  of 
this  set  afterwards  became  a  standard  of  reference.  The  angles 
were  measured  with  repeating-theodolites,  and  azimuth  was 
determined  at  five  principal  stations  by  measuring  the  angle 
between  another  station  and  the  sun,  mornings  and  even- 
ings. Latitudes  were  computed  from  zenith-distance  observa- 
tions at  the  termini  and  at  three  intermediate  points.  A  com- 
mission was  appointed  to  review  all  the  calculations :  they 
combined  this  arc  with  the  Peruvian,  deducing  the  length  of  a 
quadrant  whose  legalized  fractional  part  is  the  present  metre. 

Nouet,  while  astronomer  to  the  French  expedition  to  Africa 
in  1798,  measured  a  short  arc,  from  which  he  found  a  degree  to 
be  56,880  toises.  The  disagreement  between  the  computed 
and  observed  azimuths  obtained  by  Maupertuis — amounting  to 
34*  in  the  terminal  line — caused  considerable  suspicion  to  attach 
to  the  entire  work.  The  Stockholm  Academy  of  Sciences  de- 
cided to  have  the  stations  reoccupied,  and  consequently,  in 
1801,  sent  Svanberg.  Palander,  and  two  others  to  Lapland  for 
that  purpose.  They  did  not  recover  all  of  the  previously  oc- 
cupied stations,  nor  did  they  use  the  same  terminal  points,  but 
deduced  as  an  independent  value  for  a  degree  57,196  toises. 

Major  Lambton  measured  an  arc  of  i°  33'  56"  in  India  in  a 
mean  latitude  of  12°  N.  in  1802.  After  his  death,  in  1805,  it 
was  continued  by  Colonel  Everest  with  such  vigor  that  by  1825 
an  arc  of  16°  was  completed. 

The  French  gave  the  English  an  impetus  to  push  forward 


HISTORIC  SKETCH.  1 5 

geodetic  work  by  their  co-operation  in  the  connection  already 
referred  to,  so  that  while  in  England  a  trigonometric  survey  was 
being  prosecuted,  the  requisite  care  was  bestowed  upon  it  to 
make  it  of  value  in  degree-determinations.  From  1783  to  1800 
this  survey  was  under  the  direction  of  General  Roy.  Mudge 
continued  the  triangulation  for  two  years,  completing  an  arc  of 
2°  50',  from  which  he  found  for  the  length  of  a  degree  in  lati- 
tude 53°,  57,017  toises,  and  in  51°,  57,108  toises;  therefore  the 
degrees  shorten  towards  the  pole. 

Mechain  wished  to  carry  his  arc  south  of  Barcelona  to  the 
Balearic  Isles,  but  was  prevented  by  his  unfortunate  death. 
However,  the  energetic  mathematicians  who  made  that  period 
of  the  French  history  so  brilliant  would  not  allow  such  a  fea- 
sible project  to  remain  incomplete.  So  Biot  and  Arago  spent 
two  years,  beginning  in  1806,  in  extending  the  triangulation 
from  Mt.  Mongo,  on  the  coast  of  Valencia,  to  Formentera,  giv- 
ing a  complete  arc  of  12°  22'  13*44. 

The  latitude  of  Formentera  was  determined  from  nearly 
4000  observations  on  a  and  ft  Ursae  Minoris,  but  owing  to  the 
fact  that  they  were  all  made  on  stars  on  one  side  of  the  zenith, 
erroneous  star-places  would  introduce  serious  errors  in  the  re- 
sulting latitude,  as  demonstrated  by  Biot  in  1825,  when  he  ob- 
tained for  that  station  a  latitude  differing  by  9"  from  the  first. 
The  length  of  a  degree  as  published  in  1821  was  57,027  toises 
in  latitude  45°  N.  Bessel,  using  the  corrected  latitude  of  For- 
mentera, found  56,964  toises ;  and  in  1841  Puissant  discovered 
another  error  which  changed  the  degree's  length  to  57,032 
toises.  In  the  reduction  of  this  work  the  principal  of  least 
squares  was  used  for  the  first  time  in  adjusting  the  triangulation 
in  conformity  with  the  geometric  conditions,  as  will  be  explained 
in  a  future  chapter. 

The  errors  already  referred  to  in  the  reduction  of  this  work 
show  the  fallacy  of  accepting  any  determination  of  the  earth's 
quadrant  as  an  unvarying  quantity  from  which  a  standard,  if 


1 6  GEODETIC  OPERATIONS. 

lost  or  destroyed,  could  be  definitely  restored  with  a  length 
identical  with  the  previous  one.  Even  if  the  earth  be  perfectly 
fixed  and  stable  in  its  size  and  shape,  of  which  there  is  great 
doubt,  and  the  ten-millionth  part  of  a  quadrant  always  the 
same,  the  uncertainties  in  obtaining  the  same  value  for  this 
quadrant  twice  in  succession  outweigh  the  utility  of  the  plan 
and  the  majesty  of  its  conception.  This  is  not  intended  as  an 
argument  against  the  decimal  feature,  or  the  readiness  with 
which  units  of  weight  can  be  obtained  from  those  of  volume. 
In  this  respect  the  metric  system  is  superior  to  all  others  now 
in  use,  and  these  advantages  alone  warrant  its  universal  adop- 
tion, while  the  fixity  of  the  standards  preserved  by  the  Inter- 
national Bureau  of  Weights  and  Measures  is  sufficiently  certain 
to  dispel  all  doubts  as  to  the  change  of  length  of  the  metre, 
without  feeling  the  necessity  of  frequently  comparing  it  with 
a  physical  law  or  mass  supposed  to  be  immutable. 

Prussia  began  geodetic  work  in  1802  with  the  measurement 
of  a  base-line  near  Seeburg  by  von  Zach.  This  line  was  care- 
fully measured  and  the  end-points  fixedly  marked  by  inclosing 
in  masonry  iron  cannons  with  the  mouth  upwards.  In  the 
mouth  a  brass  cylinder  was  fastened  by  having  lead  run  around 
it ;  the  cross-lines  on  the  upper  surface  of  the  cylinders  denoted 
the  end  of  the  line.  The  triangulation  began  in  1805,  but  was 
stopped  by  the  war  with  France  in  1806,  although  Gotha,  the 
province  in  which  the  work  was  being  prosecuted,  remained 
neutral.  After  the  battle  of  Jena  the  people  of  Gotha,  fear- 
ing that  the  French  would  not  regard  their  neutrality  lasting, 
especially  if  they  should  be  suspected  of  harboring  concealed 
weapons,  caused  these  cannons  to  be  dug  out  and  carefully 
hid,  thus  sacrificing  some  accurate  work  to  allay  a  foolish 
fear. 

Under  Napoleon  I.  the  importance  of  faithful  maps  for  war 
purposes  at  least  was  keenly  felt,  and  to  secure  men  trained 
for  the  preparation  of  such  maps  the  Ingenieur  Corps  was  or- 


HISTORIC  SKETCH.  \"J 

ganized,  also  the  Ecole  Polytechnique  and  the  Ecole  Speciale 
de  Ge"ode"sie.  The  basis  of  an  accurate  cartographic  survey 
must  be  a  triangulation,  and  degree-measurements  had  such  a 
strong  hold  upon  the  mathematicians  that  the  advisability  of 
giving  to  the  triangulation  the  requisite  accuracy  to  make  it 
useful  for  such  determinations  was  never  questioned. 

Switzerland  and  Italy  were  to  join  their  work  to  that  of 
France,  to  give  an  arc  of  parallel  from  the  Atlantic  Ocean  to 
the  Adriatic  Sea.  This  was  begun  in  1811,  and  continued  by 
one  or  more  of  the  countries  until  its  completion  in  1832,  giv- 
ing an  arc  of  12°  59'  4".  Owing  to  serious  discrepancies  be- 
tween the  observed  and  computed  values,  this  work  received 
but  little  credit.  In  one  instance  the  difference  in  azimuth 
was  49". 5 5,  and  in  longitude  the  difference  between  the  geo- 
detic and  astronomic  was  51". 29. 

The  French  expedition  to  Lapland  for  the  purpose  of  an  arc- 
measurement  incited  the  first  astronomer  of  the  St.  Petersburg 
Academy,  De  1'Isle,  to  make  a  similar  determination  in  Russia. 
In  1737  he  measured  a  base-line  on  the  ice  between  Kronstadt 
and  Peterhof,  and  occupied  several  stations  during  that  and 
the  two  following  years.  However,  it  came  to  an  end  very 
abruptly  without  leaving  any  definite  results  by  which  to  re- 
member it. 

The  first  geodetic  work  in  Russia  that  deserves  the  name 
was  begun  in  1817  under  the  patronage  of  Alexander!.,  with 
Colonel  Tenner  and  Director  Struve  at  the  head.  Tenner 
began  in  the  province  of  Wilma  and  continued  until  1827,  by 
which  time  he  had  completed  an  arc  of  4^°,  using  a  base 
measured  with  an  apparatus  of  his  own  devising,  consisting  of 
two  parallel  bars  of  iron  firmly  fastened  together.  The  angles 
were  read  on  a  i6-inch  repeating  theodolite.  Struve  did  not 
receive  his  instruments  until  1821,  but  in  the  ten  years  follow- 
ing he  finished  an  arc  of  3^°. 

There  was  now  a  gap  of  about  5^°  between  the  Russian  and 


,8  GEODETIC  OPERATIONS. 

the  Lapland  arcs  which  it  was  desired  to  close  up.  In  this 
work  Struve  was  assisted  by  Argelander.  They  measured  a 
check-base  with  Struve's  apparatus,  completing  the  entire  task 
in  1844.  In  the  mean  time  Tenner  had  added  3°  25'  to  his  arc. 
Just  here  it  might  be  of  interest  to  remark  that  Bessel  had 
communicated  to  Tenner  his  discussion  regarding  the  figure 
and  size  of  the  earth.  This  was  appended  to  Tenner's  manu- 
script record  and  placed  in  the  care  of  the  St.  Petersburg 
Academy  in  1834,  three  years  before  it  was  published  by  Bes- 
sel in  the  Astronomische  Nachrichten,  No.  333. 

Permission  was  obtained  from  the  Swedish  authorities  to 
continue  this  arc  across  Norway  and  Sweden.  This  also  was 
placed  under  the  direction  of  Struve,  with  the  assistance  of 
Selander  and  Hansteen.  The  former  finished  his  share  of  the 
triangulation  with  a  measured  base  in  1850.  Hansteen  com- 
pleted the  Norwegian  portion,  checking  on  a  base  of  1155 
toises.  The  Russian  parties,  together  with  their  co-laborers, 
by  1855  had  completed  a  meridional  arc  of  25°  2O/9/'.29,  ex- 
tending from  the  Danube  to  the  North  Sea.  Of  this  there 
were  two  great  divisions — the  Russian,  with  8  bases  and  224 
principal  triangles  and  9  latitude-determinations;  and  the 
Scandinavian,  with  2  bases,  33  principal  triangles,  and  4  astro- 
nomic stations.  Prior  to  1821  the  principle  of  repetition  was 
exclusively  used  on  horizontal  circles  in  its  original  form. 
Struve  then  decided  that  the  periodic  errors  noticed  when  the 
simple  method  of  repetition  was  employed  could  be  partially 
eliminated  by  reversing  the  direction  of  rotation  ;  but  he  soon 
abandoned  this,  and  in  1822  began  to  measure  angles  a  number 
of  times  on  different  parts  of  the  circle. 

The  test  of  the  accuracy  of  this  work  is  in  the  difference  in 
the  lengths  of  junction-lines  as  computed  from  different  bases. 
From  an  examination  of  ten  of  these  differences,  I  have  found 
that  the  average  is  0.1718  toise,  with  0.0179  as  the  minimum 
and  0.4764  for  a  maximum.  The  values  found  for  a  degree 


HISTORIC  SKETCH.  19 

were:  57,092  toises  in  latitude  53" 20',  57,116  in  55°  34',  57,121 
in  56°  32,  56,956  in  57°  28',  and  57,125  in  59°  14'.  The  utility 
of  this  arc  for  degree-measurements  is  not  proportionate  to  its 
immensity,  because  of  the  fewness  of  the  astronomic  deter- 
minations— only  one  in  every  two  degrees  of  amplitude. 

General  von  Muffling  in  1818  connected  the  Observatory  of 
Seeburg  with  Dunkirk,  and  determined  the  amplitude  of  the 
arc  by  measuring  the  difference  in  time  between  the  stations 
two  by  two.  This  was  done  by  recording  in  local  time  the  ex- 
act instant  at  which  a  powder-flash  set  off  at  one  station  at  a 
known  local  time  was  seen  at  the  other.  The  amplitude  of 
this  arc,  embracing  8  determinations  of  this  kind  in  its  chain, 
was  8°2i/  1 8". 

Between  1818  and  1823  Colonel  Bonne  connected  Brest  with 
Strasburg,  with  a  base  near  Plouescat.  It  is  interesting  to 
note  that  in  this  work  angles  were  measured  at  night,  using 
as  a  signal  a  light  placed  in  the  focus  of  a  parabolic  reflec- 
tor. Differences  of  longitude  were  determined  by  powder- 
flashes. 

Gauss  began  the  trigonometric  survey  of  Hanover  in  1820, 
measuring  an  arc  of  2°  57',  from  which  he  found  for  a  degree 
57,126  toises  in  the  same  latitude  in  which  Mudge  in  England 
obtained  for  a  degree  57,016  toises,  and  Musschenbroeck,  in 
Holland,  57,033  toises.  It  was  while  engaged  upon  this  work 
that  Gauss  first  used  the  heliotrope  that  has  since  borne  his 
name. 

Schumacher  at  the  same  time  commenced  the  Danish  trian- 
gulation  with  the  advice  and  assistance  of  Struve.  His  arc 
of  i°  31' 53"  gave  for  a  degree  57,092  toises  in  latitude  54° 

8'  13". 

In  1821  Schwerd  concluded  from  his  measurement  of  the 
Speyer  base  that  a  short  line  most  carefully  measured  would 
give  as  good  results  as  a  longer  one  on  which  the  same  time 
and  labor  would  be  expended.  From  his  base  of  859.44  M.  he 


20  GEODETIC  OPERATIONS. 

computed  the  length  of  Lammle's  base  of  19,795.289  M.,  giving 
a  difference  of  only  0.0697  M. 

Colonel  Everest  was  appointed  to  succeed  Colonel  Lambton 
in  the  direction  of  the  great  trigonometric  survey  of  India  in 
1823.  During  the  following  seven  years  he  measured  three 
bases  with  the  Colby  apparatus  as  checks  to  the  triangulation 
which  he  extended  from  18°  3'  to  24°  f.  To  Colonel  Everest 
is  due  the  credit  of  introducing  greater  care  in  all  the  linear 
and  angular  determinations.  In  the  latter  he  employed 
the  method  of  directions  in  greater  number  than  did  his  pred- 
ecessors. 

In  1831  Bessel  and  Baeyer  undertook  a  scheme  of  triangula- 
tion that  was  to  unite  the  chains  of  France,  Hanover,  Den- 
mark, Prussia,  and  Bavaria  with  that  of  Russia,  and  at  the 
same  time  serve  for  degree-measurements.  It  was  oblique,  so 
that,  by  determining  the  direction  and  amplitude,  degrees  of 
longitude  as  well  as  latitude  could  be  found.  The  base-line 
near  Fuchsburg  was  measured  with  a  slightly  modified  form 
of  the  Borda  apparatus  now  known  as  Bessel's  apparatus,  of 
which  there  is  now  an  exact  copy  in  use  in  the  Landes  Trian- 
gulation of  Prussia.  The  length  of  this  base  was  934.993 
toises  when  reduced  to  sea-level.  The  ends  were  marked  by  a 
pier  of  masonry  inclosing  a  granite  block,  in  whose  top  was  set 
a  brass  cylinder  carrying  cross-lines  indicating  the  end  of  the 
base.  Just  above  this  was  built  a  hollow  brick  column  high 
enough  for  the  theodolite  support,  with  a  larger  square  stone 
for  a  cap-stone.  In  the  centre  of  this  there  was  a  cylinder 
coaxial  with  the  one  below,  so  that  the  instrument  could  be 
placed  immediately  over  the  termini  of  the  base.  The  theodo- 
lites had  12-  and  1 5-inch  circles,  read  by  verniers,  and  the  angles 
were  read  by  fixing  the  zeros  coincident,  and  then  turning  to 
each  signal  in  succession  with  verniers  read  and  recorded  for 
each.  After  completing  the  series,  the  signals  were  observed 
in  inverse  order,  the  means  of  the  two  readings  giving  a  set  of 


HISTORIC  SKETCH.  21 

directions.  The  zero  would  then  be  shifted  to  another  position, 
and  all  the  signals  sighted  both  in  direct  and  inverted  order, 
until  a  desired  number  of  sets  were  secured.  The  method  of 
reduction  is  given  on  page  99. 

Two  kinds  of  signals  were  used ;  one  consisted  of  a  hemi- 
sphere of  polished  copper  placed  with  its  axis  vertically  over  the 
centre  of  the  station.  The  sun  shining  on  this  gave  to  the  ob- 
server a  bright  point,  but  not  in  a  line  joining  the  centres  of 
the  stations  observing  and  observed  upon  ;  consequently  a  cor- 
rection for  phase,  as  explained  on  page  144  had  to  be  applied. 
The  other  form  consisted  of  a  board  about  two  feet  square, 
painted  white  with  a  black  vertical  stripe  ten  inches  wide  down 
the  centre.  This  board  was  attached  to  an  axis  made  to  coin- 
cide with  the  centre  of  the  station,  so  as^to  permit  the  board 
to  be  turned  in  a  direction  perpendicular  to  the  line  of  sight  as 
different  stations  were  being  occupied. 

The  astronomic  determinations  were  made  at  three  stations 
with  the  greatest  possible  care ;  while  the  reduction  of  the  tri- 
angulation  was  a  monument  to  the  methods  devised  by  Gauss 
for  treating  all  auxiliary  angles  as  aids  in  finding  the  most 
probable  corrections  to  be  applied  to  those  angles  absolutely 
needed  in  the  computation.  The  amplitude  of  the  arc  was 
i°  3O/28".97.  Using  the  two  parts  into  which  the  arc  was  di- 
vided by  Konigsberg,  the  difference  between  the  terminal 
points  taken  as  a  whole,  and  the  sum  of  the  two  parts  was  only 
0.973  toise,  which  is  an  evidence  of  the  great  accuracy  attained 
in  this  work.  The  report  of  this  triangulation  was  published 
in  Gradmessung  in  Ostpreussen,  und  ihre  Verbindung  mit  Preus- 
sischen  und  Russischcn  Dreiecksketten,  Berlin,  1838;  and  while 
now  nearly  half  a  century  has  elapsed  since  its  appearance,  not 
only  its  influence  is  still  felt,  but  the  operations  then  for  the 
first  time  described  are  now  in  use. 

There  is  not  a  geodesist  of  the  present  time  who  is  not  in- 
debted to  this  work  for  information  as  well  as  assistance,  and 


22  GEODETIC  OPERATIONS. 

as  long  as  exact  science  receives  attention  men  will  turn  to 
this  fountain-head.  My  greatest  inspiration  comes  from  two 
sources both  perhaps  sentimental,  but  none  the  less  effica- 
cious. My  copy  of  the  above  book  was  presented  to  Jacobi 
by  Bessel,  as  shown  by  the  latter's  superscription.  This  is  be- 
fore me  in  reality ;  the  other  remains  in  memory  as  the  cordial 
greetings  and  encouragement  of  Baeyer,  with  whom  I  worked 
in  the  Geodetic  Institute. 

From  1843  to  1861  Sir  A.  Waugh,  who  succeeded  Sir  George 
Everest,  added  nearly  8000  miles  to  the  Indian  chains.  After 
him  came  General  Walker's  administration,  and  during  the 
following  thirteen  years  he  completed  5500  miles  of  triangle 
chains,  occupied  55  azimuth  stations,  and  determined  89 
latitudes. 

In  this  work  the  triangle  sides  are  from  15  to  60  miles  in 
length.  In  those  cases  where  it  was  necessary  to  elevate  the 
instrument  masonry  towers  were  erected,  some  as  high  as  50 
feet.  Luminous  signals  were  used — heliotropes  by  day,  and 
Argand  lamps  at  night.  The  amplitude  of  the  greatest  Indian 
arc  is  23°  49'  23"-54,  but  its  exact  value  has  been  questioned, 
owing  to  the  uncertainties  of  the  effect  of  local  attractions  in  the 
neighborhood  of  the  Himalayas  upon  the  latitudes  and  azi- 
muths, as  well  as  the  negative  attraction  along  the  shore  of  the 
Indian  Ocean  as  pointed  out  by  Archdeacon  Pratt.  When 
the  computed  effects  of  these  attractions  are  applied,  there  is 
still  a  discrepancy. 

A  meridional  arc  of  about  30°  has  been  completed,  but  owing 
to  the  impracticability  of  ascertaining  the  difference  of  longi- 
tudes its  amplitude  is  not  accepted  as  sufficiently  accurate  to 
warrant  its  use  in  degree-determinations. 

The  purpose  of  this  great  trigonometric  survey  was  to  fur-, 
nish  a  basis  for  topographic  maps ;  consequently  the  chains  of 
primary  triangles  are  parallel  at  such  a  distance- apart  as  to 
allow  the  intervening  country  to  be  easily  covered  with 


HISTORIC  SKETCH.  2$ 

secondary  triangles  with  the  primaries  for  checks  on  each  side 
of  the  chasm.  There  are  24  chains  running  north  and  south, 
and  7  east  and  west. 

Between  1847  and  I^5I  the  Russian  chain  was  connected 
with  the  Austrian,  having  12  sides  in  common  ;  the  greatest 
discrepancy  being  o.ioi  toise,  and  the  least  o.oi  toise. 

About  the  same  time  the  junction  of  the  Lombardy  and 
Swiss  chains  showed  a  difference  of  0.31  and  0.34  metre. 

In  1848  the  astronomer  Maclear  revised  Lacaille's  Good 
Hope  arc,  extending  it  to  an  amplitude  of  3  degrees,  from 
which  he  deduced  for  I  degree,  in  latitude  35°  43',  56,932.5 
toises.  Comparing  this  with  the  French  arc  in  approximately 
the  same  northern  latitude,  we  find  a  difference  of  only  48 
toises  in  a  degree. 

In  1831  Borden  devised  a  base-apparatus  with  which  he 
measured  a  base  and  began  a  triangulation  over  the  State  of 
Massachusetts,  making  the  commencement  of  geodetic  work 
in  the  United  States.  Borden  read  his  angles  with  a  12-inch 
theodolite,  using  the  method  of  repetition.  Latitudes  were 
determined  from  circumpolar  altitude  observations  at  24 
points. 

Recently  many  of  his  stations  have  been  re-occupied,  intro- 
ducing greater  care  in  all  features  of  the  work  and  affording  a 
check  on  Borden's  results.  Comparing  the  two  sets  of  values 
for  the  geographical  positions  of  the  stations  that  are  common, 
it  appears  that  there  is  a  systematic  increase  in  the  errors,  being 
the  greatest  in  the  eastern  part  of  the  State,  that  being  the 
furthest  from  the  base-line.  The  average  discrepancy  in  the 
linear  determination  is  1:11000,  or  somewhat  less  than  6  inches 
in  a  mile. 

The  United  States  Coast  Survey,  organized  in  1807,  had 
primarily  for  its  object  the  survey  of  the  coast,  but  this  ne- 
cessitated a  carefully  executed  triangulation  of  long  sides  to 
check  the  short  triangle  sides  whose  terminal  stations  were 


24  GEODETIC  OPERATIONS. 

sufficiently  near  one  another  for  the  coast  topography  and  off- 
shore hydrography.  It  soon  became  apparent  that  but  little, 
if  any,  additional  care  was  needed  to  secure  sufficient  accuracy 
to  make  this  trigonometric  work  a  contribution  to  geodesy. 
By  1867  an  arc  of  3°  23'  was  completed,  extending  from  Farm- 
ington,  Maine,  to  Nantucket,  with  two  base-lines,  seven  latitude 
stations,  and  ten  determinations  of  azimuth. 

Summing  the  six  arcs  into  which  the  whole  naturally  divides 
itself,  it  was  found  that  a  degree  in  latitude  43°  and  longitude 
70°  20'  was  111,096  metres,  or  57,000.5  toises. 

By  1876  the  Pamlico-Chesapeake  arc  of  4°  3i'-5  was  com- 
pleted, embracing  in  its  chain  of  triangles  six  bases  and  fourteen 
astronomic  stations.  The  latitude  of  each  of  these  stations  was 
computed  from  the  one  nearest  the  middle  of  the  arc,  and  the 
difference  between  this  and  the  observed  values,  called  station- 
error,  attributed  to  local  deflection.  This  in  no  case  exceeded 
3^  seconds;  and  in  general  it  was  in  accord  with  a  uniform 
law  disclosed  by  the  geology  of  the  country  over  which  the  arc 
extends. 

From  an  elaborate  discussion  of  the  sources  of  error  in  this 
arc,  Mr.  Schott  concludes  that  the  probable  error  in  its  length 
is  not  in  excess  of  3^  metres.  The  length  of  a  degree  in  lati- 
tude 37°  16'  and  longitude  76°  08' is  56,999.9101568. 

The  triangulation  is  being  continued  southward,  and  in  a 
very  short  time  it  is  hoped  that  the  entire  possible  arc  of  22° 
will  be  reduced  and  the  results  announced.  An  arc  of  parallel 
is  also  under  way,  keeping  close  to  the  39th. 

Of  this  great  arc  of  49°  about  three  fourths  is  completed. 
This  is  the  longest  arc  that  can  anywhere  be  measured  under 
the  auspices  of  a  single  country.  Consequently,  considering 
the  great  advantage  to  be  derived  from  perfect  harmony  of 
methods,  it  is  no  wonder  that  scientists  in  all  parts  of  the 
world  are  anxiously  awaiting  the  completion  of  this  important 
work.  Also,  when  done,  it  will  be  well  done.  The  high  stand- 


HISTORIC  SKETCH.  2$ 

ard  of  excellence  introduced  into  this  service  at  its  beginning 
f"  makes  the  first  results  comparable  with  the  most  recent. 

In  1857  Struve  advocated  the  project  of  connecting  the 
triangulations  of  Russia,  Prussia,  Belgium,  and  England,  giving 
an  arc  of  69°  along  the  52d  parallel.  Bessel  had  already  made 
the  Prussian -Russian  connection,  and  in  1861  England  and  Bel- 
gium joined  with  tolerable  success,  finding  in  their  common 
lines  discrepancies  amounting  to  an  inch  in  a  mile. 

The  Prussian  and  Belgium  chains  are  not  yet  satisfactorily 
united  ;  neither  are  the  longitudes  determined. 

While  a  topographic  map  of  Italy  was  begun  in  1815,  no 
special  interest  was  taken  in  geodesy  until  1861,  except  in 
rendering  some  slight  assistance  in  that  part  of  the  French  and 
Austrian  triangulation  that  overlapped.  In  this  year  Italy  re- 
sponded to  the  suggestion  of  Baeyer,  adopted  by  the  Prussian 
Government,  to  form  an  association  of  the  European  powers  to 
measure  a  meridional  and  a  parallel  arc. 

The  Italian  Commission  was  formed  in  1865,  and  at  once 
elaborated  plans  for  future  work.  It  was  decided  to  have  six 
chains  of  triangles,  and  for  every  twenty  or  twenty-five  a  care- 
fully measured  base  ;  also  to  connect  Sicily  with  Africa  ;  direc- 
tion-theodolites of  10-  and  12-inch  circles  to  be  used.  The  base- 
apparatus  with  which  the  first  three  bases  were  measured  was 
of  the  Bessel  pattern.  The  base  of  Undine  was  measured  with 
the  Austrian,  and  the  next  two  with  a  Bessel  equipped  with 
reading-microscopes  for  reading  the  divisions  on  the  glass 
wedges. 

The  numerous  observatories  are  connected  with  the  trigono- 
metric stations,  and  one  or  two  are  to  be  erected  in  the  merid- 
ian of  the  arc  to  determine  its  deflection. 

The  geodetic  work  in  Spain  began  with  the  measurement  of 
the  Madridejos  base  in  1858.  The  apparatus  used  in  this  work 
was  specially  designed  for  it,  and  the  precision  introduced 
into  the  measurement  of  the  base,  as  well  as  in  the  depend- 


26  GEODETIC  OPERATIONS. 

ing  triangulation,  has  given  to  the  Spanish  work  great  confi- 
dence. 

This  is  especially  fortunate,  as  it  will  form  an  important  link 
in  the  chain  extending  from  the  north  of  Scotland  into  Africa, 
and  in  the  oblique  chain  from  Lapland  to  the  same  point.  In 
addition  to  the  central  base  first  measured,  three  others  were 
found  necessary  to  check  the  system. 

The  general  plan  resembles  that  pursued  in  the  India  Survey 
in  having  parallel  chains  at  such  a  distance  from  one  another 
that  the  intervening  country  can  be  readily  filled  in  with  sec- 
ondary triangles  for  the  topographic  purposes. 

There  are  three  of  these  meridional  chains  with  amplitudes 
of  about  six,  seven,  and  seven  and  a  half  degrees,  and  an  arc  of 
parallel  of  twelve  degrees. 

Likewise  the  Swedish  coast-triangulation  was  begun  in  1758 
for  the  purpose  of  checking  the  coast-charts,  and  in  1812  an- 
other triangulation  embracing  fifty  stations  and  five  base-lines, 
measured  with  wooden  rods,  was  started  for  a  similar  end. 
However,  it  was  not  until  the  announcement  of  Bessel's  results 
that  Sweden  took  an  active  interest  in  accurate  work. 

In  1839  the  Alvaren  base  was  measured  with  Bessel's  appara- 
tus, and  again  in  the  following  year  with  the  same  bars,  giving 
a  difference  of  0.0145  metre  in  the  two  results. 

So  far  the  work  was  purely  cartographic,  and  it  was  the  in- 
fluence of  Baeyer  that  caused  a  partial  transformation  in  the 
methods,  making  them  conformable  to  the  system  of  the  Per- 
manent Commission  for  European  Degree-measurements. 

Three  bases  have  been  measured  with  a  modified  Struve  ap- 
paratus, giving  excellent  results ;  in  one  instance  the  difference 
between  the  two  measurements  being  only  0.0029  metre,  and 
twenty-nine  stations  occupied,  using  Reichenbach  and  Repsold 
theodolites. 

Under  the  auspices  of  this  commission  the  following  coun- 
tries are  prosecuting  geodetic  work :  Austria,  Bavaria,  Belgium, 


HISTORIC  SKETCH.  27 

France,  Hesse,  Holland,  Italy,  Portugal,  Prussia,  Russia,  Sax- 
ony, Spain,  Switzerland,  and  Wiirtemberg. 


LITERATURE    OF    THE    HISTORY    OF    GEODESY. 

Verhandlungen  der  allgemeinen  Conferenz  der  Europaischen 
Gradmessung. 

Roberts,  Figure  of  the  Earth,  Van  Nostrand's  Engineering 
Magazine,  vol.  32,  pp.  228-242. 

Comstock,  Notes  on  European  Surveys. 

Baeyer,  Ueber  die  Grosse  und  Figur  der  Erde. 

Posch,  Geschichte  und  System  der  Breitengradmessungen. 

Merriman,  Figure  of  the  Earth. 

Baily,  Histoire  de  I'Astronomie. 

Wolf,  Geschichte  der  Vermessungen  in  der  Schweiz. 

Clarke,  Geodesy. 

Westphal,  Basisapparate  und  Basismessungen. 

Klein,  Zweck  und  Aufgabe  der  Europaischen  Gradmessung. 


GEODETIC  OPERATIONS. 


CHAPTER  II. 

INSTRUMENTS    AND    METHODS    OF    OBSERVATION. 

THE  perfection  of  an  instrument  is  the  result  of  corrected 
defects,  and  in  the  development  of  geodesy  or  degree-meas- ' 
urements  improved  methods  were  closely  followed  by  better 
instruments.     So  that  while  discussing  the  progressive  steps 
of  one,  the  other  cannot  be  wholly  neglected. 

For  the  uncultured  peoples,  distances  can  be  given  with  suf- 
ficient accuracy  as  so  many  days'  journey,  and  nothing  but  the 
necessity  to  carry  on  record  some  measured  magnitude  would 
call  for  a  unit  that  could  be  readily  attained.  The  first  such 
unit  of  which  there  is  any  authentic  information  is  the  Chaldean 
mile,  which  was  equal  to  4000  steps  of  a  camel ;  the  next  was 
the  Olympian  race-course,  giving  to  the  Greeks  their  unit — 
the  stadium.  The  rods  with  which  the  Arabians  measured  the 
two  degrees  already  mentioned — known  as  the  black  ell — • 
have  been  lost,  and  not  even  their  equivalent  length  known. 

Fernel,  in  using  the  wagon-wheel  for  a  measuring  unit,  found 
it  quite  constant  in  length  and  of  a  kind  easily  applied, — advan- 
tages that  are  appreciated  to  this  day  by  topographers,  who 
frequently  measure  meander  lines  by  having  a  cyclometer  at- 
tached to  a  wheel  of  a  vehicle. 

When  Snellius  devised  the  method  of  triangulation  there 
were  needed  two  forms  of  instruments — one  for  linear  measure- 
ments, and  another  for  angle-determinations.  At  this  time 
angles  were  measured  with  a  quadrant  to  which  sights  were 
attached  ;  a  rectangle  with  an  alidade  and  sights  pivoted  to  one 
of  the  longer  sides,  the  other  being  divided  into  degrees ;  a 


INSTRUMENTS  AND  METHODS  OF  OBSERVATION.       2Q 

square  with  the  alidade  in  one  corner  and  all  four  sides  gradu- 
ated ;  a  compass  with  sights ;  a  semicircle  with  alidade  or 
compass  at  the  centre.  Also  for  navigators  there  was  the  as- 
trolabe, an  instrument  devised  by  Hipparchus  for  measuring 
the  altitude  of  the  sun  or  a  star. 

Defects  in  graduation  were  early  detected,  and  efforts  to 
avoid  them  made  by  increasing  the  radius  of  the  sector,  the 
smallest  used  by  the  first  astronomers  being  of  6  and  7  feet 
radius  ;  and  it  is  said  that  a  pupil  of  Tycho  Brahe  constructed 
a  sector  of  14  feet  radius;  while  Humboldt  says  the  Arabian 
astronomers  occasionally  employed  quadrants  of  180  feet  ra- 
dius. In  the  case  of  large  circles,  or  parts  of  circles,  the  divi- 
sions that  could  be  distinguished  would  be  so  numerous  as  to 
render  the  labor  of  dividing  very  great,  and  the  intermediate 
approximation  uncertain. 

Nunez,  a  Portuguese,  in  1542  devised  a  means  of  estimating 
a  value  smaller  than  the  unit  of  division.  He  had  about  his 
quadrant  several  concentric  circular  arcs,  each  having  one  divi- 
sion less  than  the  next  outer,  so  that  the  difference  between  an 
outer  and  an  inner  division  was  one  divided  by  the  number  of 
parts  into  which  the  outer  was  divided.  This  differs  from  our 
present  vernier,  first  used  by  Petrus  Vernierus  in  1631,  in  which 
the  auxiliary  arc  is  short  and  is  carried  around  with  the  zero- 
point. 

A  great  impetus  was  given  to  applied  mathematics  by  the 
construction  of  logarithmic  tables,  according  to  the  formulae  of 
Napier  (1550-1617),  and  Briggs  (15 56-1630),  especially  in  facili- 
tating trigonometric  computations,  which  had  now  become  the 
basis  of  degree-measurements. 

The  first  person  to  use  an  entire  circle  instead  of  a  part  was 
Roemer  in  1672,  who  deserves  our  thanks  for  having  invented 
the  transit  also.  Auzout  in  1666  made  the  first  micrometer, 
and  Picard  was  the  first  to  apply  it,  and  a  telescope  with  cross- 
wires,  to  an  angle-reading  instrument.  The  results  obtained 


30  GEODETIC  OPERATIONS. 

with  this  instrument  were  so  satisfactory  that  Cassini  used  it 
in  his  great  triangulation  begun  eleven  years  later.  The  angles 
in  Peru  were  measured  with  quadrants  of  21,  24,  30  and  36 
inches  radius,  each  provided  with  one  micrometer.  These 
gave  very  fair  results — the  maximum  error  in  closure  of  a  tri- 
angle being  12  seconds,  spherical  excess  not  considered.  This 
would  give  an  error  of  one  unit  in  5000  in  the  length  of  a  de- 
pending line — a  value  ten  times  better  than  any  obtained  dur- 
ing the  preceding  century. 

With  such  close  reading  of  angles  the  discrepancies  between 
measured  and  computed  lines  were  quite  naturally  attributed 
to  the  unit  of  measure,  the  method  of  its  use,  or  its  comparison 
with  a  standard.  As  early  as  the  Peruvian  work  the  uncer- 
tainty in  the  varying  length  of  wooden  rods  because  of  damp- 
ness, and  of  metal  rods  on  account  of  heat,  was  appreciated ; 
and  in  the  measurement  of  these  bases  an  approximate  average 
of  13°  R.  was  assumed  for  the  mean  temperature.  This  hap- 
pened to  be  the  temperature  at  which  the  field  standards  had 
been  compared  with  the  copy  before  leaving  Paris,  hence  the 
reason  for  legalizing  this  temperature  for  that  at  which  the 
toise  of  Peru  is  a  standard. 

In  1752  Mayer  announced  the  advantages  to  be  derived 
from  repeating  angles,  and  a  repeating-circle  was  constructed 
upon  this  principle  by  Borda  in  1785,  for  the  connection  of  the 
French  and  English  work.  The  first  dividing  engine  was  made 
by  Ramsden  in  1763,  and  a  second  improved  one  in  1773, 
which  did  such  good  work  that  his  circles  soon  became  deserv- 
edly famous.  In  1783  this  maker  furnished  an  instrument  to 
the  English  party  engaged  upon  the  work  just  mentioned, 
this  was  the  first  to  be  called  theodolite.  It  had  a  circle  three 
feet  in  diameter,  divided  into  ten-minute  spaces,  read  by  two 
reading  micrometer  microscopes.  One  turn  of  the  micrometer- 
screw  was  equal  to  one  minute,  and  the  head  was  divided  into 
sixty  parts,  so  that  a  direct  reading  to  a  single  second  could 


INSTRUMENTS  AND  METHODS  OF  OBSERVATION.       31 

be  made,  and  to  a  decimal  by  approximation.  It  was  also 
provided  with  a  vertical  circle  of  10.5  inches  diameter,  read  by 
two  micrometers  to  three  seconds.  The  success  attained  in 
the  use  of  this  instrument,  giving  a  maximum  error  of  closure 
of  three  seconds,  was  regarded  as  truly  phenomenal. 

Reichenbach  began  the  manufacture  of  instruments,  in 
Munich,  in  1804,  °f  such  a  high  grade  of  workmanship  that  it 
was  soon  considered  unnecessary  to  send  to  Paris  or  London  in 
order  to  secure  the  best.  He  fortunately  furnished  Struve  with 
a  theodolite,  putting  a  good  instrument  in  the  hands  of  one  of 
the  most  skilful  observers  who  has  ever  lived,  which  contrib- 
uted no  little  to  his  reputation.  His  circles  were  almost  wholly 
repeaters,  a  class  of  instruments  exclusively  used  on  the  Con- 
tinent, but  not  at  all  in  England. 

Littrow,  at  the  Observatory  of  Vienna,  was  the  first  to  aban- 
don the  method  of  repetition,  in  1819;  and  Struve,  in  1822,  was 
the  next  to  follow. 

The  inconvenience  attending  the  use  of  large  circles  was 
very  great,  besides  the  irregularities  produced  from  flexure  on 
account  of  unequal  distribution  of  supports.  This  led  to  the 
attempt  to  make  a  smaller  circle  with  good  graduation,  and 
reading-microscopes.  This  end  was  achieved  by  Repsold,  who 
made  a  ten-inch  theodolite  for  Schumacher  in  1839,  with  which 
it  was  definitely  demonstrated  that  as  good  results  could  be  se- 
cured with  a  ten  or  a  twelve  inch  instrument  as  with  a  larger  one, 
and  with  less  expenditure  of  time  and  labor,  not  considering 
the  difference  in  the  first  cost.  So  that  now  we  find  the  effort 
heretofore  spent  in  constructing  enormous  circles  given  to  per- 
fecting the  graduation,  and,  while  using  the  instrument,  to  pro- 
tect the  circle  from  sudden  or  unequal  changes  of  temperature. 
Mr.  Saegmuller's  principle  of  bisection  in  dividing  a  circle 
keeps  the  errors  of  graduation  within  small  limits,  and  the  new 
dividing  engines  leave  but  little  to  be  desired  in  the  construc- 
tion of  theodolites. 


32  GEODETIC  OPERATIONS. 

In  England  and  India  eighteen-inch  circles  are  now  used 
in  place  of  those  of  twice  that  size  formerly  employed. 
Struve  had  a  thirteen-inch  theodolite.  In  the  U.  S.  Coast  and 
Geodetic  Survey  the  large  instruments  have  given  way  to  those 
of  twelve  inches.  In  Spain  twelve-  and  fourteen-inch  circles  are 
found  to  be  the  best,  while  the  excellent  work  of  the  U.  S.  Lake 
Survey  was  done  with  theodolites  having  circles  of  twenty  and 
fourteen  inches  in  diameter — the  latter  having  the  preference. 

To  describe  the  various  forms  of  theodolites  now  in  use 
would  necessitate  a  number  of  illustrations,  and  in  the  end  be 
tedious  and  unprofitable;  the  same  general  features  being  com- 
mon to  all,  they  only  will  be  referred  to.  The  end  sought  in 
the  construction  of  theodolites  is  to  get  an  instrument  with 
parts  sufficiently  light  to  insure  requisite  stability,  with  circles 
large  enough  to  allow  close  readings,  with  the  telescopic  axis 
concentric  with  the  circle,  a  reliable  means  for  subdividing  the 
divisions  on  the  circle,  and  a  circle  so  graduated  as  to  be  free 
from  errors,  or  to  have  them  according  to  a  law  readily  dis- 
tinguished and  easily  allowed  for.  While  every  one  concedes 
that  the  foregoing  requisites  are  imperative,  in  respect  to  some 
there  is  a  great  difference  of  opinion  as  to  when  they  are  at- 
tained. 

The  illustration  appended  shows  an  eight-  to  twelve-inch 
theodolite  of  the  form  suggested  by  the  experience  of  the  skilled 
officers  of  the  U.  S.  Coast  and  Geodetic  Survey.  In  its  construc- 
tion hard  metal  is  employed,  and  as  few  parts  used  as  possible. 
The  frame  is  made  of  hollow  or  ribbed  pieces  in  that  shape 
that  gives  the  greatest  strength  for  the  material.  The  bearings 
are  conical ;  clamps  of  a  kind  that  avoid  travelling  motion  ;  the 
circle  is  solid,  and  of  a  conical  shape  to  prevent  flexure.  The 
focal  distance  is  diminished  so  as  to  admit  of  reversal  of  tele- 
scope without  removing  it  from  its  supports,  and  the  optical 
power  is  increased  to  insure  precision  in  bisecting  a  signal. 
They  are  made  as  nearly  symmetrical  as  possible,  and  when 


INSTRUMENTS  AND  METHODS  OF  OBSERVATION. 


there  is  no  counterpoise  provided,  one  of  the  proper  weight  is 
put  in  place.  They  are  furnished  with  three  foot-screws  for 
levelling,  resting  in  grooves  converging  towards  the  centre. 
Sometimes  a  circular  level  is  set  in  the  lowest  part  of  the 
branching  supports,  and  in  other  cases  a  single  tubular  level  is 
made  use  of.  The  optical  axis  is  marked  by  having  in  the 
principal  focus  spider-lines  called  a  reticule,  or  a  piece  of  very 
thin  glass  on  which  fine  lines  are  etched.  The  arrangement  of 
the  lines  is  various,  the  forms  depicted  in  the  annexed  cut  be- 
ing the  ones  most  frequently  found. 


The  instrument  shown  in  Fig.  I  is  one  of  directions  in  which 
the  circle  is  shifted  for  new  positions.  With  a  repeater  the 
only  difference  is  the  addition  of  a  slow-motion  screw  to  move 
the  entire  instrument  in  accordance  with  the  method  of  repe- 
tition as  explained  on  page  98. 

The  adjustments  of  a  theodolite  must  be  carefully  attended 
to  and  frequently  tested.  They  may  be  described  in  general 
as  follow : 

To  Adjust  the  Levels. — When  the  tripod  or  stand  is  placed  in 
a  stable  condition  and  the  instrument  mounted,  bring  it  into  a 
level  position,  as  indicated  by  the  level,  by  turning  the  foot- 
screws.  Turn  the  instrument  180  degrees,  correct  any  defect, 
— one  half  by  means  of  the  screws  attached  to  the  level,  and 
the  rest  by  the  foot-screws.  Place  the  instrument  in  its  first 
position,  repeat  the  corrections  as  before  until  no  deviation  is 
noticed  when  the  circle  is  turned.  If  there  is  a  second  level,  it 
is  to  be  adjusted  in  the  same  manner. 
z 


34 


GEODETIC  OPERATIONS. 


FIG.  i. 


INSTRUMENTS  AND  METHODS  OF  OBSERVATION.       35 

To  Adjust  the  Spider-lines  of  the  Telescope.— (i)  Place  the 
threads  in  the  focus  of  the  eye-piece,  point  to  a  suspended 
plumb-line  when  the  air  is  still,  and  see  if  the  vertical  thread 
coincides  with  the  plumb-line.  If  there  is  any  deflection,  loosen 
the  four  screws  holding  the  diaphragm  and  move  it  gently  till 
there  is  a  coincidence,  then  tighten  the  screws  and  verify.  (2) 
If  the  level  is  correct,  place  the  circle  in  a  horizontal  position 
and  sight  to  some  clearly  defined  object ;  move  the  instrument 
sideways  by  means  of  the  tangent  screw  and  notice  if  the  hori- 
zontal thread  traverses  the  point  throughout  its  entire  length, 
if  not,  correct  as  in  the  above  case. 

To  Adjust  the  Line  of  Collimation  of  the  Telescope. — When 
the  horizontal  axis  of  the  telescope  can  be  reversed,  point  the 
instrument  to  some  clearly  denned  object,  then  reverse  the 
telescope  and  see  if  the  pointing  is  good.  If  not,  half  the  dif- 
ference is  to  be  corrected  in  the  pointing  and  the  other  half 
by  moving  the  entire  diaphragm  to  the  right  or  left,  as  the  case 
may  be.  Continue  this  course  until  the  pointing  remains  per- 
fect after  reversal.  If  the  instrument  does  not  admit  of  this 
reversal,  it  must  be  turned  in  its  Y's  ;  and  if  the  reading  is  more 
or  less  than  180  degrees  from  the  first  reading,  correct  as  be- 
fore, until  there  is  just  180  degrees  between  the  readings  before 
and  after  reversal. 

The  horizontality  of  the  axis  of  the  telescope  is  tested  by 
placing  on  the  axis  a  portable  level  that  is  in  good  adjustment. 
If  a  defect  is  apparent,  it  must  be  corrected  entirely  by  raising 
or  lowering  the  movable  end. 

After  completing  these  adjustments,  it  is  well  to  repeat  the 
tests  to  see  if  any  have  been  disturbed  while  the  other  ad- 
justments were  in  progress.  When  large  instruments  with 
reading-microscopes  are  used,  the  corrections  for  runs  and 
eccentricity  must  be  determined.  The  former  can  be  readily 
ascertained  as  follows:  Turn  the  micrometer  in  the  direction 
of  the  increasing  numbers  on  its  head  till  the  movable  cross- 


36  GEODETIC  OPERATIONS. 

wire  bisects  the  first  five-minute  space  ;  call  the  reading  a. 
Reverse  the  motion  and  continue  to  the  preceding  five-minute 
space  ;  call  this  b.  Suppose 

a  =  45°  40'  +  4'  46".4,  b  =  45°  40'  -f  4'  44"- 2, 

r  =  a  -  b  =  +  2". 2,  m  =  ^~~  =  A-'  45"-3- 

Since  the  five-minute  space  contains  300  seconds,  the  correction 
to  a  =  r  .a  -f-  300  =  —  2".i  ;  correction  to  b  =  r(b  —  300) 
_i_  300  =  -j-  ".i  i  ;  correction  to  m  —  \(a  -\-  b  —  300)^  -=-  300 
=  —  o".88,  in  which  a  and  b  represent  minutes  and  seconds  of 
the  above  readings.  The  corrected  reading  is  therefore, 

45°  44' 45 ".3  -  .88  =45°44'44".42. 

Occasionally  the  average  error  of  runs  is  determined  and  a 
table  computed  from  the  formula  just  given  for  a  -f-  b  from  5 


d 


to  10  seconds.  But  in  very  accurate  work  the  correction  for 
runs  is  made  for  each  reading  by  recording  the  two  micrometer- 
readings  just  mentioned  for  each  pointing.  They  are  recorded 
as  forward  and  backward,  as  seen  on  page  101. 

The  eccentricity  is  owing  to  the  centre  of  the  axis  carrying 
the  telescope  not  coinciding  with  the  centre  of  the  graduated 
circle.  As  each  point  on  the  plate  carrying  the  telescope  must 


INSTRUMENTS  AND  METHODS  OF  OBSERVATION.       37 

return  to  its  former  position  after  each  complete  revolution, 
there  must  be  a  point  at  which  there  is  a  maximum  deflection 
as  well  as  a  point  at  which  there  is  no  deflection,  and  at  the 
same  time  the  intermediate  positions  have  eccentric  errors  be- 
tween these  limits ;  therefore  it  is  necessary  to  examine  the 
whole  circle.  This  can  be  done  in  connection  with  an  exami- 
nation of  the  two  verniers.  The  difference  in  the  reading  of  the 
two  verniers  may,  however,  be  due  to  other  causes:  the  con- 
stant angular  distance  between  them  may  be  more  or  less  than 
1 80  degrees,  or  it  may  be  owing  to  errors  of  graduation,  or 
errors  of  reading,  or  to  the  eccentricity  referred  to. 

Let       c  be  the  centre  of  the  limb, 
m  that  of  the  telescope, 
6  =  angle  amb, 
0'  =  angle  deb, 
E  —  the  difference,  or  error, 
e  =  cm  =  the  linear  eccentricity, 
oo  =  dcm, 

r  =  radius  of  the  circle, 
d  =  cdm, 
b  =  cbm, 
dm  =  0  +  b  —  Q'+d;  therefore,  E—Q-Q'^d-b. 


As  cm  is  never  very  large,  we  can  put  mb  =  r :  in  the  tri- 
igle  cdi 
we  have 


angle  cdm,  we  have  sin  d  =  -  sin  GO,  and  in  the  triangle  bcm, 


sin  b  —  -  sin  bcm  =  -  sin  (GO  —  6'). 

• 

Also,  since  d  and  b  are  small,  we  can  write  for  sin  b,  b.  sin  i", 
and  for  sin  d,  t/.sin  \" ,  so  that  we  have, 


38  GEODETIC  OPERATIONS. 


;7[sin  oo  -  sin  (GO  - 


By  expanding  sin  (oo  —  0'),  and  putting  for  the  entire  angles 
their  values  in  terms  of  the  half-angles,  we  find, 

E  =  '7sin  *6'  '  C°S   Q?  ~ 


This  expression  is  made  up  of  two  factors,  and  becomes  o 
when  either  factor  becomes  o,  as  e  —  o,  or  cos  (oo  —  £#')  =  o, 
that  is,  when  oo  —  $0'  —  90°,  or  6'  =  200  —  180°, 

Therefore  when  the  verniers  are  180°  apart,  the  errors  of 
eccentricity  are  eliminated.  Likewise  E  is  a  maximum  when 
cos  (oo  —  $0'}  =  -|-i,  that  is,  when  GJ  —  \ff  =  o,  or  2oo  ==  6'. 

In  accord  with  the  principle  that  errors  of  eccentricity  are 
avoided  when  the  angle  is  read  from  two  points  180°  apart, 
circles  are  provided  with  two  verniers  that  distance  from  each 
other.  Instead  of  verniers,  however,  we  may  have  two  micro- 
scopes. 

The  practical  difficulty  of  placing  the  zero-points  just  180° 
apart  makes  it  necessary  to  examine  each  circle  to  see  what 
the  angular  distance  between  them  is.  This  is  best  accom- 
plished by  setting  one  vernier,  say  A,  on  each  10°  mark,  and 
reading  and  recording  vernier  B.  If  a  represent  the  amount 
by  which  the  angular  distance  differs  from  180°,  and  b  the  effect 
of  eccentricity  on  this  distance,  we  will  have  B—  A—  180° 
-\-a-\-b,  and  when  the  verniers  change  places  b  will  have  a 
contrary  effect,  so  that  B  —A  =  180°  +  a  —  b  ;  therefore  if  we 
take  the  mean  of  the  differences  B  —  A  for  positions  that  are 
just  180°  apart,  we  will  have  the  angular  distance  unaffected  by 
eccentricity.  We  so  arrange  our  readings  as  to  have  on  the 
same  line  those  that  are  180°  apart.  We  also  place  under  B 
—  A  the  first  difference,  and  on  the  same  line  the  second  dif- 
ference, the  mean  will  be  the  average  of  the  two,  or  180°  -+-  a, 


INSTRUMENTS  AND  METHODS  OF  OBSERVATION.       39 


and  the  average  of  these  means  will  be  the  mean  distance  be- 
tween the  verniers. 


FIRST. 

SECOND. 

B  -A. 

A. 

B. 

A. 

B. 

ISt. 

2d. 

Mean. 

0°  00'  00" 

1  80°  00'  05" 

1  80°  00'  00" 

0°  00'  00" 

t  5" 

O 

+  2".  5 

IO 

10 

190 

05 

+  10 

+  5 

+  7.5 

20 

05 

200 

00 

+  5 

o 

-M  .5 

30 

IO 

210 

05 

+  IO 

4-  5 

4-7  -5 

40 

55 

2  2O 

oo 

-  5 

o 

-  2  .5 

50 

00 

230 

05 

o 

4-  5 

+  2  .5 

60 

05 

240 

IO 

+  5 

+  10 

+  7  -5 

70 

05 

250 

05 

+  5 

+  5 

+  5 

80 

10 

260 

oo 

+  10 

o 

4-5 

90 

05 

270 

10 

4-  5 

+10  +  7  .5 

100 

oo 

280 

55 

o 

-  5  -  2  .5 

no 

55 

290 

00 

-  5 

0 

-2.5 

1  20 

55 

300 

05 

-  5 

+  5 

o 

130 

05 

310 

05 

4-  5 

T  5 

4-5 

140 
150 

05 
05 

320 

330 

55 
oo 

£•! 

—  5 
o 

o 

4-2  .5 

160 

05 

340 

oo 

4-  5 

o 

4-2  .5 

170 

05 

350 

05 

4-  5 

+  5 

4-5 

Therefore  the  angular  distance  =  180°  +  3".!.     Mean  =  3".!. 

Now,  knowing  the  angular  distance  between  the  two  verniers, 
the  difference  between  it  and  the  mean  of  B  —  A  will  be  the 
errors  of  eccentricity  and  graduation,  or  b -\- g. 

Angle  dcA  =  m-\-A,  therefore  A  — 
dcA  —  m.  If  we  call  afthe  reading  on  the 
limb  which  is  on  the  line  of  no  eccentricity, 
that  is  on  the  line  drawn  through  the 
centre  of  motion  and  centre  of  graduation, 
and  n  any  angle  read  by  the  verniers, 
then  n  —  d  will  be  the  angle  between  the 
vernier  and  line  of  no  eccentricity,  or  dcA. 
In  the  triangle  Acm,  sin  Acm  :  sin  A  :: 
Am  :  cm,  but  sin  Acm  =  sin  dcA  =  sin  (n  —  d),  and  Am  =  r, 
nearly,  making  these  substitutions: 


FIG.  3. 


4O  GEODETIC  OPERATIONS. 

.  e.sin(n—d) 

sin  (n  —  d):smA::r'.e,        or        sin  A  =  --  . 

A  being  small,  we  can  put  for  sin  A,  A  .  sin  i",  and  the  angu- 
lar value  for  e  to  radius  r,  e.s'm  l"  ;  then  write  for  A  in 
seconds,  A  =  e.s'm  (n  —  d),  and  for  the  two  verniers,  6  = 
2e.  sin  («  —  d\  A  reading  b'  at  180°  from  the  former  will 
have  the  same  error,  but  with  an  opposite  sign,  b'  =  —  2e 
.  sin  (n  —  d}.  If  we  tabulate  the  differences  between  the 
mean  in  our  first  table  and  the  various  readings  for  B  —  A, 
placing  on  the  same  line  those  that  differ  by  180°  from  one  an- 
other, they  should  be  equal  with  opposite  signs  were  it  not  for 
errors  of  graduation;  let  these  differences  be  D  and  D',  then 
=/>,  and*'  +*•=/>', 


2e  sin  (n  — 

—  2e  sin  (n  —  d)-\-g  =  D' 


Subtracting, 

4/?  sin  (»  -  d)  =  D  -  D',        2e  sin  («  -  d)  =  %(D  -  D'}  —  b, 

or  a  value  for  b  freed  from  errors  of  graduation.  This  will  give 
18  equations  involving  e  and  n. 
Placing  8  =  $(D-  D'\  we  have  ; 


?,  =  2e sin  (10°  -d)  =  2*(sin  10°  cosd—  cos  10°  sin  d}\ 
rw  =  2* sin  (170°  -  d)  =  2t(s'm  170°  cos  d  —  cos  170°  sin  d). 


INSTRUMENTS  AND  METHODS  OF  OBSERVATION.       41 

Professor  Hilgard's  method  for  solving  these  equations  with 
respect  to  2e  cos  d  and  2e  sin  d,  by  least  squares,  is  to  multiply 
each  equation  through  by  cos  «,  and  sum  the  resulting  equa- 
tions ;  then  each  through  by  sin  «,  and  sum  the  results  :  this 
will  give  us  two  normal  equations  of  this  form  ;  after  factoring 
2e  cos  d,  and  —  2e  sin  d, 

[#!  sin  o°  -f-  $1  sin  10°  .  .  .  Slt  sin  170°] 

=  2^cos^[sinso°  +  sin*  iO°  .  .  .  sin"  I/O0] 

—  2e  sin  */[sin  o°  cos  o°+sin  10°  cos  io°+.  .  .  sin  170°  cos  170°]; 

[tf,  cos  o°  +  d,  cos  10°  ...  d18  cos  170°] 

=  2e  cos  d  [cos  o°  sin  o°  .  .  .  cos  170°  sin  170°] 
—  2e  sin  d?[cos8  o°  +  cos2  ioa  .  .  .  cos8  170°] 

sin  o°  cos  o°  =  o,  also  for  sin  10°  cos  10°  we  can  put  £sin  20° 
and  so  on  with  all  the  products  of  sines  times  cosines  ;  and  we 
find  that  this  will  give  us  pairs  of  angles  that  make  up  360°, 
whose  sines  are  equal  but  with  opposite  algebraic  signs,  so  the 
products  reduce  to  zero.  Again,  we  can  arrange  the  second 
powers  so  that  all  angles  above  90°  can  be  written  90°  -f-  n  ; 
sin*  (90  -f-  n)  =  cos2  n,  this  added  to  sin2  n  =  I,  for  example  ; 
sin8  o°  =  o,  sin8  10°  +  sin8  100°  =  sin2  10°  +  sin2  (90°  +  10°)  = 
sin2  10°  +  cos2  10°  =  i. 
This  will  give  us  half  as  many  unities  as  we  have  terms  less 

N 
two  for  the  pairs,  and  sin2  90°  =  i  gives  us  9  =  —  .     The  nor- 

mal equations  will  then  reduce  to 


sin  «)  =  Ne  cos  d, 
cos  n)  =  —  Ne  sin  d  ; 


GEODETIC  OPERATIONS. 


tt. 

First 
6  +  f- 

Second 

ISt-2d        t 

sin  ». 

cosn. 

S  sin  n. 

5  cos  n. 

a 

0° 

•4-  2. 

-   2.9 

4     -5 

O.OO 

I.OO 

o".oo 

4  2".  50 

IO 

4  7- 

+   2.1 

4     -5 

•17 

.98 

4°    -43 

42    -45 

20 

4  2. 

—   2.9 

+     -5 

•34 

•  94 

4  o  .85 

42    .35 

30 

4° 

4  7- 

—  7. 

+   2.1 
—    2.9 

4     -5 
—     -5 

•  50 
.64 

.87 
.76 

+  i    .25 
—  I   .60 

42    .17 
-  i    .90 

50 
60 

-   2. 
4-  2. 

-j-2.1 

4  7-1 

—     -5 
-     -5 

•  76 
.87 

.64 
•  50 

-  I  .90 

-  2     .I? 

-  i    .60 
-  i    .25 

70 

T  2- 

4  2.1 

•94 

•34 

o    .00 

o   .00 

80 

-  2.9 

+      .0 

.98 

•17 

44    -90 

40  .85 

9° 

4  2. 

4  7-1 

—       5 

I.OO 

.00 

—   2     .50 

o    .00 

100 

—   2. 

—  7-9 

4     -5 

.98 

-    -17 

42    -45 

-o    .43 

no 

-  7- 

—  2.9 

-    .5 

•94 

—   -34 

-2    .35 

40  .85 

120 

—  7> 

4  2.1 

-  5-0 

.87 

-   -50 

-4    -35 

+  2     .50 

130 
140 

--  2. 
--   2. 

4  2.1 

-  7.9 

o 
4  5-0 

.76 
.64 

-   .64 
-   -76 

o    .00 
43    .20 

0     .00 

-3    -80 

150 

160 

--  2. 
--  2. 

—  2.9 
-  2.9 

42.5 

4  2.5 

•50 
•34 

-   -87 
-   -94 

tl  -25 
o    .85 

-   2     .I? 
-   2     .35 

170 

4  2. 

4  2.1 

o 

•17 

-   .98 

o   .00 

o    .00 

2(8  sin  n)  — 

40  .31 

2(8  cos  n)   = 

40     .I? 

tan  </=—/  =  tan  151°  15'  40"; 


e  — 


0.17 


iSsin  151°  15'  40 


77  =  —  O".O2. 


The  line  of  no  eccentricity  is  that  passing  through  151°  15' 
40";  the  sign  of  e  being  minus,  we  know  that  the  centre  of  mo- 
tion is  in  the  opposite  direction  from  the  centre  of  graduation 
towards  the  reading  d.  In  this  case  it  is  too  small  to  be  con- 
sidered. To  determine  the  error  of  graduation,  we  compute 
the  values  of  2e  sin  (n  —  d}  =  b;  subtracting  these  results 
from  those  in  the  last  table  marked  ^-f-^in  the  first  column, 
we  will  have  g.  It  is  necessary  to  compute  b  for  every  10° 
space,  only  up  to  180°,  since  b  has  the  same  value  for  180° 
-{-  n  that  it  has  for  n,  with  the  opposite  sign,  then  subtract 
these  values  from  the  second  b  -\- g. 


INSTRUMENTS  AND  METHODS  OF  OBSERVATION.       43 


n. 

n-d. 

it  sin  (»  —  d}. 

n. 

'* 

«. 

f- 

O° 
IO 

-    151° 
—    141 

+  0.017 
+      -025 

—    0° 
10 

4-  2".  083 

4-7  -075 

1  80° 
I90 

-    2.883 
4-    2.125 

2O 
30 

—    121 

4-    -030 

+      -034 

20 

30 

+2     .070 

+  7    -066 

20O 

210 

—    2.870 
+    2.134 

40 

—    Ill 

+      -037 

40 

-  7    -937 

2  2O 

-    2.863 

50 

—    IOI 

4-    -038 

50 

-  2    .938 

230 

4-   2.138 

60 

—     91 

4-   .040 

60 

+2     .060 

240 

+    7-I40 

70 

-     81 

+   -038 

70 

4-2     .062 

250 

+    2.138 

80 

—     71 

+   -037 

80 

+  7    -063 

260 

-    2.863 

90 

-     61 

4-   -034 

90 

+2     .066 

270 

+    7-134 

IOO 

—     51 

4-    .030 

IOO 

-  2     .930 

280 

-    7-870 

no 

-     41 

4-    .026 

no 

-  7    .926 

290 

-    2.874 

120 

4"       -O2O 

1  20 

—  7    .920 

300 

-\-   2.I2O 

130 
140 

—       21 
—       II 

4-  .014 
4-    -007 

130 
140 

-f-  2     .086 
+  2       093 

310 
320 

4-   2.II4 

-  7  893 

150 
1  60 
170 

—      OI 
+      09 
+      19 

-j-     .000 

-    .006 
—    .013 

150 

160 
170 

4-2     .100 

+2     .106 
4-2     .113 

330 
340 
350 

—    2  .900 
—    2.906 

4-  2.087 

.The  sum  of  the  squares  of  the  36  values  for  g  give  714.7617, 
therefore  the  probable  error  in  any  one  is  A  /  — -  = 

±  4".6 ;  this  divided  by  the  square  root  of  two  gives  the  prob- 
able error  of  the  reading  of  one  vernier,  owing  to  errors  of  grad- 
uation and  accidental  errors  of  reading  =  ±  3".2. 

If  an  angle  is  the  mean  of  five  repetitions,  the  probable  error 
of  the  average  will  be  one  fifth  of  3//.2  =  ±  o".64. 

If  the  effect  of  eccentricity  be  considerable,  the  correction  to 
each  angle  should  be  computed  by  the  equation  b  =  2e  sin 
(»  —  d\  The  probable  error  in  graduation  and  reading  is 
used  only  in  computing  the  probable  error  in  a  chain  of  tri- 
angles, as  will  be  seen  later.  If  the  instrument  has  two  read- 
ing-microscopes the  procedure  is  essentially  the  same,  but  dif- 
fers slightly  when  there  are  three.  In  this  case  every  5°  or  10° 
space  can  be  examined  and  the  three  microscopes  read ;  as  be- 
fore, we  shall  call  the  reading  of  the  zero-point  «,  and  the 
microscopes  A,  B,  and  C. 


44 


GEODETIC  OPERATIONS. 


n. 

*+!*>•. 

«  +  240*. 

0°  00'  00" 

+  01" 

+  02" 

120  00  01 

oo 

—  01 

Sum 
One  third 

240  oo  02 

—  04 

oo 

+  03 

-  02.6 

-  00.9 

—  03 
+  03-3 

-j-  OI.I 

+°' 

—  0.6 

—  O.2 

=  +  I,  average  =  +  0.3. 


The  first  line  gives  the  readings  when  the  zero-point  is  n, 
the  order  of  the  microscopes  is  A,  B,  and  C\  in  the  next,  zero 
is  at  n  -J-  120°,  and  the  order  is  C,  A,  and  B;  in  the  third,  zero 
is  at  «  +  240°,  ancl  tne  order  is  B,  C,  and  A.  The  fourth  line 
contains  the  sums,  and  the  continuation  the  average  ;  and  by 
subtracting  the  sums  from  this  average  we  have  the  fifth  line 
containing  three  times  the  errors  of  trisection  at  this  point. 

Eccentricity  is  first  determined  :  "  Suppose  acn,  fin,  and  yn  be 
the  observed  errors  of  trisection  corresponding  to  n,  n-\-  120°, 
and  n  +  240°,  also  [o^cos  »],  [/?«cos(«-|-  120°)],  [<*„  sin  »]  .  .  . 
etc.,  be  the  sums  of  all  the  an  cos  n,  an  sin  n,  etc.,  then  dt  the 
line  of  no  eccentricity, 

=  _  [an  cos  n\  +  [/?,  cos  (n  +  120°)]  +  Q«  cos  (n  +  240°)] 
[an  sin  »]  +  [/?„  sin  («  +  120°)]  +  [yn  sin  (»  +  240°)] 

also  e" 

_  __  [^  sin  «]  4-  [/?„  sin  («+  120°)]  -f  \yn  sin  («  -f  240°)] 


where  ^V  =  number  of  trisections. 

"  The  correction  for  eccentricity  is  b  —  e  sin  (n  —  d\  then  if 
<**',  P*',  Yn  —  errors  of  trisections  freed  from  errors  of  eccen- 
tricity, we  will  have  : 


INSTRUMENTS  AND  METHODS  OF  OBSERVATION.       45 

an'  =  an  —  e  sin  (n  —  d) ; 

/?„'  =  /3n  -  e  sin  (n  +  120°  —  d] ; 

Yn  =  Yn  —  e  sin  («  +  240°  —  d). 

Knowing  «»',  /?„',  ynf,  the  residuals  are  squared,  and  the  prob- 
able error  of  graduation  and  reading  found  as  in  the  preceding 
case." 

Considering  that  the  determination  of  latitude,  longitude, 
and  azimuth  forms  a  part  of  practical  astronomy,  the  only  in- 
struments that  remain  to  be  described  are  the  base-apparatus 
and  heliotrope.  The  former  is  referred  to  in  the  chapter  on 
base-measuring,  and  the  latter  can  be  dismissed  with  a  few 
words. 

The  first  heliotrope  was  used  by  Gauss  in  1820.  It  was 
somewhat  complicated,  consisting  of  a  mirror  attached  to  the 
objective  end  of  a  small  telescope.  This  mirror  had  a  narrow 
middle-section  at  right  angles  to  the  rest  of  it ;  this  was  in- 
tended to  reflect  light  into  the  tube,  while  the  remainder  re- 
flected the  sun's  rays  upon  the  object  towards  which  the  tele- 
scope was  pointed.  Bessel  devised  a  much  simpler  form  that  is 
still  in  use  in  Prussia.  It  has  a  small  mirror,  with  two  motions, 
fastened  to  one  end  of  a  narrow  strip  of  board,  while  at  the 
other  end  there  is  a  short  tube  whose  height  above  the  board 
is  the  same  as  the  axis  of  the  mirror.  In  this  tube  cross-wires 
are  stretched,  and  a  shutter  can  be  dropped  over  the  end  op- 
posite the  mirror.  To  use  it,  one  fastens  the  screw  that  is  at- 
tached to  one  end  in  a  suitable  support  and  then  by  means 
of  a  levelling  screw  at  the  other  end,  raises  or  lowers  that 
end  until  the  centre  of  the  mirror,  the  cross-wires  and  the 
object  towards  which  the  light  is  to  be  reflected  are  in  line. 
The  mirror  is  then  turned  so  that  the  shadow  of  the  cross- 
wires  falls  upon  their  counterpart  that  is  marked  on  the  shutter 
when  the  light  can  be  seen  at  the  desired  point.  Perhaps  the 
most  convenient  of  all  is  the  heliotrope  that  finds  employment 


46  GEODETIC  OPERATIONS. 

in  the  U.  S.  Coast  and  Geodetic  Survey.  It  can  be  seen  in 
Fig.  4.  First  of  all,  there  is  a  low-power  telescope  provided 
with  a  screw  for  attachment  to  a  tree  or  signal.  On  one  end 
of  the  tube  is  a  fixed  ring  of  convenient  diameter,  say  one 
and  a  half  inches,  while  at  the  other  end  is  a  mirror  of  two 
inches  in  diameter,  and  at  an  intermediate  point,  nearer  the 
mirror,  is  another  ring  of  the  same  height  and  size  as  the  other, 
but  clamped  to  the  tube,  admitting  of  a  motion  around  it. 

To  describe  its  use  we  will  suppose  it  in  adjustment.  After 
having  screwed  it  to  a  post,  the  telescope  is  turned  until  the 
cross-wires  approximately  coincide  with  the  point  to  which  the 
light  is  to  be  shown ;  then  turn  the  mirror  so  that  the  shadow 
of  the  nearer  ring  exactly  coincides  with  the  other  ring.  Then 
as  the  earth  revolving  places  the  sun  in  a  different  relative 
position,  it  will  be  necessary  to  continually  move  the  glass  in 
order  to  keep  the  shadow  of  the  back  ring  on  the  front  one. 
If  the  sun  is  behind  the  heliotrope  an  additional  mirror  will  be 
needed  to  throw  the  light  upon  the  glass. 

To  effect  the  adjustment,  it  is  necessary  to  have  in  the  con- 
struction the  centres  of  the  rings  and  the  mirror  at  the  same 
distance  from  the  optical  axis  of  the  telescope.  Bisect  some 
clearly  defined  point,  then  sight  over  the  tops  of  the  mirror 
and  rings,  turning  the  movable  one  until  they  are  all  in  line 
with  the  object  bisected  by  the  telescope.  Owing  to  the  large 
diameter  of  the  sun,  a  slight  error  in  adjusting  will  not  affect 
the  successful  use  of  this  kind  of  a  heliotrope. 

When  the  observed  and  observing  stations  are  within  twenty 
miles  of  one  another,  the  light  spot  may  be  too  large  to  be 
•  easily  bisected ;  then  it  is  best  to  place  between  the  glass  and 
rings  a  colored  glass  (orange  is  preferable),  so  as  to  reduce  the 
light  as  seen  to  a  mere  spot.  A  code  of  signals  can  be  adopted 
and  messages  exchanged  between  observer  and  heliotroper, 
such  as  "  Correct  your  pointing,"  "  Stop  for  the  day,"  "  Set 
on  new  station,"  "  Too  much  light,"  "  Not  enough  light,"  by 


INSTRUMENTS  AND  METHODS  OF  OBSERVATION.       tf 

cutting  off  the  light  with  a  hat  or  small  screen  ;  a  long  stoppage 
standing  for  a  dash,  and  a  short  one  for  a  dot,  when  the  words 
can  be  spelled  out  by  the  Morse  code. 

The  maximum  distance  at  which  a  heliotropic  signal  can  be 
seen  depends  upon  the  condition  of  the  atmosphere.  Perhaps 
the  greatest  was  on  the  "  Davidson  quadrilateral,"  where  a 
light  was  seen  at  a  station  192  miles  away. 

A  very  convenient  form  of  heliotrope,  especially  for  recon- 
noissance,  is  one  invented  by  Steinheil,  and  known  by  his  name. 
It  differs  from  all  others  in  having  only  one  mirror  and  no 


FIG.  4. 


rings,  making  it  so  simple  in  use  and  adjustment  as  to  form 
a  valuable  instrument.  The  glass  has  but  one  motion,  but 
the  frame  has  another  at  right  angles  to  it. 

As  can  be  seen  from  the  illustration,  the  entire  instrument 
can  be  attached  to  an  object  by  means  of  a  wood  screw,  and 
clamped  in  any  position  by  other  screws.  In  the  centre  of 
the  mirror  the  silvering  is  erased,  making  a  small  hole  through 
which  the  light  of  the  sun  can  pass ;  also  in  the  centre  of 
the  frame  carrying  the  mirror  there  is  an  opening  fitted  with 
a  convex  lens,  and  behind  the  lens  is  a  white  reflecting  sur- 
face— usually  chalk.  To  use  the  heliotrope,  turn  the  glass  so 
that  the  bright  point  caused  by  the  sun  shining  through  the 


48 


GEODETIC  OPERATIONS. 


hole  coincides  with  the  opening  in  the  frame.  This  will  give 
in  the  focus  of  the  lens  an  image  of  the  sun,  which  will  be  re- 
flected back-  through  the  hole  in  the  glass.  Now, 
if  the  entire  instrument  be  turned  so  as  to  bring 
this  image  upon  the  point  at  which  the  light  is 
to  be  seen,  the  rays  falling  upon  the  mirror  will 
be  reflected  in  the  same  direction. 

To  see  the  fictitious  sun,  as  the  image  is  called, 
one  must  look  through  the  hole  from  behind  the 
glass,  and  as  it  is  always  small  and  quite  indis- 
tinct, some  practice  will  be  needed  to  recognize 
it.  This  can  best  be  acquired  by  turning  the 
image  upon  the  shaded  side  of  a  house,  then  it 
will  be  seen  as  a  small  full  moon.  The  reflect- 
ing surface  can  be  moved  in  or  out  by  a  screw 
from  behind,  and  the  only  adjustment  that  is  ever  needed 
is  to  have  this  surface  at  that  distance  that  gives  the  best 
image  of  the  sun.  After  having  placed  the  heliotrope  in 
the  correct  position,  it  should  be  clamped,  and  then  the  only 
labor  is  simply  to  occasionally  turn  the  glass  so  as  to  bring 
the  bright  spot  into  coincidence  with  the  opening  in  the  frame. 
In  the  Eastern  States,  through  air  by  no  means  the  clearest,  a 
light  from  a  Steinheil  heliotrope  has  been  observed  upon  at  a 
distance  of  55  miles. 
They  are  made  by  Fauth  of  Washington. 


FIG.  5. 


104 

BA  SE-MEA  S  UREMENTS.  49 


CHAPTER  III. 

BASE-MEASUREMENTS. 

As  the  foundation  of  every  extended  scheme  of  trigonomet- 
ric surveys  must  be  a  linear  unit,  it  is  essential  that  the  length 
of  this  base  should  be  determined  with  the  utmost  degree  of 
care. 

But  the  labor  and  expense  of  measuring  a  base  of  favorable 
length  are  so  great  as  to  preclude  repeated  measurements.  In 
order,  therefore,  to  secure  results  at  all  comparable  with  the 
precision  desired,  an  apparatus  of  great  delicacy  is  needed. 
This  becomes  apparent  when  we  consider  that  an  apparatus  of 
convenient  length  is  repeated  from  one  to  two  thousand  times 
in  the  measurement  of  a  base,  and  that  even  a  small  error  in 
the  length  of  the  measuring  unit  will  be  multiplied  so  as  to 
seriously  affect  the  results. 

And  this  error  in  a  short  line  will  be  increased  proportion- 
ally in  the  computed  lengths  of  the  long  sides  of  the  appended 
triangles.  The  figure  and  magnitude  of  the  earth  are  deter- 
mined from  extended  geodetic  operations,  and  the  elements  so 
determined  are  conditionally  used  in  the  re-reduction  of  trian- 
gulation  data,  securing  in  this  way  a  more  probable  expression 
for  the  shape  of  our  planet. 

From  this  it  may  be  seen  that  all  of  our  errors  are  of  an  ac- 
cumulative character,  and  seriously  affect  the  results  unless 
fortuitously  eliminated  by  a  principle  of  compensation. 

Since  geodesy  first  received  attention,  the  subject  of  most 
important  consideration  has  been  the  construction  of  a  base- 
apparatus  that  would  secure  good  results  without  sacrificing 
4 


jo  GEODETIC  OPERATIONS. 

time  and  expense.  The  first  form  consisted  of  simple  wooden 
bars,  resting  on  stakes  previously  levelled,  and  placed  end  to 
end.  When  the  configuration  of  the  ground  made  it  necessary 
to  make  a  vertical  offset,  it  was  done  by  means  of  a  plumb-line. 
Another  form  similar  to  this  had  a  groove  cut  in  the  under 
side  to  rest  upon  a  rope  drawn  taut  from  two  stakes  of  equal 
elevation.  In  place  of  laying  the  rods  on  stakes  or  on  a  catenary 
curve,  it  was  once  found  convenient  to  place  them  on  the  ice,  as 
when  Maupertuis  measured  the  base  in  Lapland  in  1736.  This 
line  was  measured  twice,  each  time  by  a  different  party ;  the 
difference  between  the  two  results  was  four  inches.  This  was 
close  work  in  a  measurement  extending  over  a  distance  of  8.9 
miles.  The  rods  used  in  this  case  were  thirty-two  feet  long, 
made  of  fir  and  tipped  with  metal  to  prevent  wearing  by  attri- 
tion. The  Peru  base  measured  at  about  the  same  time  gave  a 
difference  of  less  than  three  inches  in  the  two  measurements 
in  a  distance  of  7.6  miles.  The  wooden  rods  were  found  to 
be  affected  by  changes  in  the  hygrometric  conditions  of  the 
atmosphere.  This  change  was  diminished  by  painting  them. 
Finally  wood  was  abandoned  as  the  material,  and  glass  tubes 
substituted.  Of  course  with  glass  there  was  a  continual  change 
in  length  due  to  expansion  or  contraction  by  thermal  varia- 
tions, that  was  not  perceptible  in  the  case  of  wood,  but  know- 
ing the  rate  of  expansion,  the  absolute  length  at  any  tempera- 
ture can  be  theoretically  computed.  The  temperature  of  each 
tube  during  the  entire  measurement  was  ascertained  by  the 
application  of  a  standard  thermometer,  and  the  length  of  the 
whole  base  was  reduced  to  a  temperature  of  62°  Fahr.  The 
difficulty  of  determining  the  temperature  of  the  tubes  was 
considerable,  since  the  thermometer  reading  gives  the  temper- 
ature of  the  mercury  in  the  thermometer,  or,  at  best,  that  of 
the  external  air,  which  will  always  differ  from  the  temperature 
of  the  measuring-bar.  In  the  case  of  a  sudden  change  of  tem- 
perature, the  thermometer  will  respond  more  quickly  than  the 


BA  SE-MEA  SUREMENTS.  5 1 

tubes,  and  its  reading  could  not  be  taken  as  the  reading  of  the 
tubes.  This  trouble  suggested  the  construction  of  an  appara- 
tus that  would  serve  to  indicate  change  in  temperature — as  a 
metallic  thermometer.  On  this  principle,  Borda  made  four  rods 
for  the  special  committee  of  the  French  Academy  in  1792. 
The  rods  were  made  of  two  strips  of  metal — one  of  platinum, 
and  the  other  of  copper  overlying  the  former.  They  were 
fastened  together  at  one  end,  but  free  at  the  other  and  through- 
out the  remaining  length.  The  copper  was  shorter  than  the 
platinum  by  about  six  inches.  It  carried  a  graduated  scale, 
moving  by  the  side  of  a  vernier  attached  to  the  platinum ;  the 
reading  of  the  scale  indicated  the  relative  lengths  of  the  two 
strips,  and  hence  the  length  and  temperature  of  the  platinum. 
The  strips  rested  upon  a  bar  of  wood — the  entire  apparatus 
being  six  French  feet  in  length.  Contact  was  made  by  a  slide, 
the  end  of  which  was  just  six  feet  from  the  opposite  end  of  the 
platinum  strip  when  the  zero-mark  on  the  slide  coincided 
with  one  on  the  end  of  the  strip  to  which  it  was  attached. 
The  rods  rested  upon  iron  tripods  with  adjusting-screws  for 
levelling,  and  the  inclination  was  ascertained  from  a  sector 
carrying  a  level.  It  is  interesting  to  note  that  the  length  of 
the  metre  was  first  determined  from  the  length  of  the  quad- 
rant computed  from  the  base  measured  with  this  apparatus. 
Borda's  compensating  apparatus  in  some  form  has  been  used 
ever  since  it  first  came  into  notice.  The  principal  varieties 
are  :  Colby,  Bache-Wurdeman,  Repsold,  Struve,  Bessel,  Hos- 
sard,  Borden,  Porro,  Reichenbach,  Baumann,  Schumacher, 
Bruhns,  Steinheil. 

In  these  varietjes — named  after  their  inventors  or  improvers 
— the  essential  features  sought  for  are : 

i.  The  terminal  points  used  as  measuring-extremities  must, 
during  the  operation,  remain  at  an  unvarying  distance  apart, 
or  the  variations  therefrom  must  admit  of  easy  and  accurate 
determination. 


52  GEODETIC  OPERATIONS. 

2.  The   distance    between  these    extremities  must  be  com- 
pared with  a  standard  unit  to  the  utmost  degree  of  accuracy, 
and  the  absolute  length  determined. 

3.  In   its    construction    provision  must  be  made  to  secure 
readiness   in    transportation,  ease   and   rapidity   in    handling, 
stability  of  supports  and  accuracy  in  ascertaining  exact  con- 
tact and  inclination. 

The  above  conditions  were  secured  in  a  great  degree  in  the 
Bache-Wurdeman  apparatus,  as  used  in  the  U.  S.  Coast  and 
Geodetic  Survey  since  1846.  The  description  given  by  Lieu- 
tenant Hunt  in  1854  will  be  found  quite  explicit.  For  the 
benefit  of  those  who  cannot  consult  the  report  which  contains 
this  description  the  following  abstract  is  given :  the  apparatus 
sent  to  the  field  consists  of  two  measuring-tubes  exactly  alike, 
each  being  packed  for  transportation  in  a  wooden  box ;  six 
trestles  for  supporting  the  tubes — three  being  fore  trestles  and 
three,  rear  trestles — each  of  which  is  packed  in  a  three-sided 
wooden  box ;  eight  or  more  iron  foot-plates  on  which  to  place 
the  trestles,  and  a  wooden  frame  is  afterwards  made  to  serve 
as  a  guide  in  laying  down  the  foot-plates  ;  a  theodolite  for 
making  the  alignment,  and  for  occasionally  referring  the  end 
of  the  tube  to  a  stake  driven  in  the  ground  for  the  purpose;  a 
standard  six-metre  bar  of  iron  in  its  wooden  case,  and  a  Sax- 
ton  pyrometer  for  effecting  a  comparison. 

The  measuring-bar  consists  of  two  parts — a  bar  of  iron  and  a 
bar  of  brass,  each  less  than  six  metres  in  length. 

These  are  supported  parallel  to  each  other ;  at  one  end  are 
so  firmly  connected  together  by  means  of  an  end-block,  in 
which  each  bar  is  mortised  and  strongly  screwed,  as  to  preserve 
at  that  point  an  unalterable  relation.  The  brass  bar,  which  has 
the  largest  cross-section,  is  sustained  on  rollers  mounted  in 
suspended  stirrups  ;  the  iron  bar  rests  on  small  rollers  which 
are  fastened  to  the  iron  bar,  and  run  on  the  brass  one.  Sup- 
porting-screws through  the  sides  of  the  stirrups  are  adjusted  to 


BASE-MEASUREMENTS.  53 

sustain  the  bars  in  place,  and  also  serve  to  rectify  them.  Thus, 
while  the  two  bars  are  relatively  fixed  at  one  end,  they  are 
elsewhere  free  to  move ;  and  hence  the  entire  expansion  and 
contraction  are  manifested  at  one  end.  The  difference  in  the 
length  of  the  two  bars  is  read  on  a  scale  attached  to  the  iron 
bar  by  means  of  a  vernier  fastened  to  the  brass  bar.  The 
scale  is  divided  into  half  millimetres,  of  which  the  vernier  indi- 
cates the  fiftieth  part,  so  that  by  means  of  a  long-focus  micro- 
scope the  difference  may  be  read  to  the  hundredth  part  of  a 
millimetre  without  opening  the  case.  Since  the  compensation 
(described  further  on)  can  be  made  correct  within  its  thirtieth 
part,  it  is  evident  that  the  true  length  of  the  compound  bars 
may  be  known  at  any  time  from  the  scale-reading,  with  an  un- 
certainty no  greater  than  the  thousandth  part  of  a  millimetre 
or  a  micron. 

The  medium  of  connection  between  the  free  ends  of  the  two 
bars  is  the  lever  of  compensation,  which  is  joined  to  the  lower 
or  brass  bar  by  a  hinge-pin,  around  which  it  turns  during 
changes  of  temperature.  A  steel  plane  on  the  end  of  the  iron 
bar  abuts  against  an  agate  knife-edge  on  the  inner  side  of  the 
lever  of  compensation.  This  lever  terminates  in  a  knife-edge, 
turned  outward  at  such  a  distance  from  the  centre-pin  and  the 
other  knife-edge  bearing,  that  the  end  edge  will  remain  un- 
moved by  equal  changes  of  temperature  in  the  two  bars.  The 
end  edge  presses  against  a  steel  face  in  a  loop  made  in  the 
sliding-rod.  This  rod  slides  in  a  frame  fastened  to  the  top  of 
the  iron  bar,  and  passes  through  a  spiral  spring,  which  acts 
with  a  constant  force  to  press  the  loop  against  the  knife-edge. 
The  outer  end  of  the  sliding-rod  bears  the  limiting  agate  plane. 
Thus  tire  end  agate  is  not  affected  in  position  by  the  expan- 
sions of  the  brass  and  iron,  acting  as  they  do  at  proportional 
distances  along  the  lever  of  compensation,  measured  from  its 
sliding-end  bearing.  The  rates  of  expansion  for  iron  and  brass 
may  safely  be  taken  as  uniform  between  the  extreme  expan- 


54  GEODETIC  OPERATIONS. 

sions  and  contractions  to  which  they  are  subject  in  practice,  and 
the  compensating  adjustment  once  made  is  permanent. 

The  stirrups  sustaining  the  rollers  on  which  the  brass  bar 
runs  are  made  fast  to  the  main  horizontal  sheet  of  the  iron 
supporting  and  stiffening  work.  This  consists  of  a  horizontal 
and  a  vertical  plate  of  boiler-iron,  joined  along  the  middle  line 
of  the  horizontal  sheet  by  two  angle-irons,  all  being  perma- 
nently riveted.  Circular  openings  are  cut  out  from  both  plates 
to  lighten  them  as  much  as  practicable.  A  continuous  iron 
tie-plate,  turned  up  in  a  trough-form,  connects  the  bottoms  of 
all  the  stirrups.  At  the  ends,  stiffening  braces  connect  the 
two  plates. 

We  now  pass  from  the  compensating  to  the  sector  end  of  the 
tube,  at  which  extremity  are  arranged  the  parts  giving  the 
readings,  and  for  adjusting  the  contacts  between  successive 
tubes  in  measuring,  thus  making  it  the  station  of  the  principal 
observer.  The  sector-end  terminates  in  a  sliding-rod,  which 
slides  through  two  upright  bars,  and  at  its  outer  end  bears  a 
blunt  agate  knife-edge,  horizontally  arranged,  which  in  measur- 
ing is  brought  to  abut  with  a  uniform  pressure  against  the 
limiting  agate  plane  of  the  compensating  end  of  the  previous 
tube.  At  its  inner  end,  this  sliding-rod  rests  against  a  cylindri- 
cal surface  on  the  upright  lever  of  contact,  so  mounted  as  at  its 
bottom  to  turn  around  a  hinge-pin.  At  top,  this  lever  rests 
against  a  tongue,  or  drop-lever,  descending  from  the  middle  of 
the  level  of  contact,  which  is  mounted  on  trunnions.*  The 
sliding-rod,  when  forced  against  the  side  of  the  lever  of  contact, 
presses  its  top  against  the  tongue  of  the  level,  and  thus  turns 
the  level  by  overcoming  a  preponderance  of  weight  given  to  its 
farther  end,  to  insure  the  contact  being  always  at  a  constant 

*  The  device  of  the  level  of  contact  is  supposed  to  be  due  to  the  elder  Repsold, 
who  applied  it  first  to  the  comparing- apparatus  used  by  Bessel,  in  constructing 
the  Prussian  standards  of  length.  A  duplicate  of  that  comparator  was  procured 
for  the  Coast  Survey,  by  F.  R.  Hassler,  Superintendent,  in  1842. 


BA  SE-MEA  SURE  MEN  TS. 


55 


pressure  between  the  agates,  the  same  force  being  always 
quired  to  bring  the  bubble  to  the 
centre.     The  arrangement  at  the 
two  ends  is  shown  in  Fig.  6. 

The  sector  is  a  solid  metal 
plate,  mounted  with  its  centre  of 
motion  in  the  line  of  the  sliding- 
rod,  and  having  its  arc  graduated 
from  a  central  zero  to  the  limits 
of  ascending  and  descending 
slopes  on  which  the  apparatus  is 
to  be  used.  A  fixed  vernier  in 
contact  with  the  arc  gives  the 
slope-readings.  A  long  level  and 
bubble-scale  are  so  attached  and 
adjusted  to  the  face  of  the  sector- 
plate  that  the  zeros  of  the  level 
and  of  the  limb  correspond  to 
the  horizontal  position  of  the 
whole  tube.  If,  then,  on  slopes, 
the  bubble  be  brought  to  the 
middle  by  raising  or  lowering  the 
arc-end  of  the  sector  (a  move- 
ment made  by  a  tangent-screw, 
whose  milled  head  projects  above 
the  tin  case  of  the  tube),  the 
vernier  will  give  the  slope  at 
which  the  tube  is  inclined,  and 
the  sloping  measure  is  readily 
reduced  to  the  horizontal  by 
means  of  a  table  prepared  for 
the  purpose.  The  level  of  con- 
tact and  the  lever  of  contact, 
with  their  appendages,  are  all  mounted  on  the  sector  and  par- 
take of  its  motions.  A  knife-edge  end  of  the  sliding-rod  presses 


56  GEODETIC  OPERATIONS. 

on  the  cylindrical  face  of  the  contact-lever,  this  cylinder  being 
concentric  with  the  sector,  and  the  sector  can  therefore  be 
turned  without  deranging  the  contact.  In  fact,  the  contacts 
are  made  with  the  sector-level  horizontal,  thus  insuring  the  ac- 
curacy of  the  contact-pressure.  The  contact-lever  is  supported 
at  bottom  by  two  braces  dropping  down  from  the  sector-plate, 
and  a  spring,  acting  on  a  pin  in  the  lever,  steadies  it  against  an 
adjusting  screw-end.  A  bracket  from  the  sector-plate  receives 
the  trunnions  of  the  contact-level.  A  small  screw  projects  from 
the  end  of  the  tube  to  clamp  or  set  the  lever  and  level  of  con- 
tact against  a  pin  in  the  sector  for  security  in  transportation. 

What  is  called  the  fine  motion,  required  for  adjusting  the 
contacts  between  the  successive  tubes,  is  produced  by  means 
of  a  compensating  rod  or  tube,  one  end  of  which  is  attached  to 
the  truss-frame  by  a  bracket  over  the  rear  trestle,  and  the  other 
receives  a  screw  terminating  in  a  projecting  milled  head.  This 
screw  turns  freely  in  a  collar,  bearing,  by  a  projecting  arm, 
against  the  cross-bar  which  joins  the  main  brass  and  iron  bars, 
and  its  nut  is  in  the  end  of  the  compensation-rod.  By  turning 
the  screw  in  one  direction,  the  bars  are  pushed  forward,  and 
the  opposite  turning  permits  a  spiral  spring,  arranged  for  the 
purpose,  to  push  back  the  system  of  bars,  which  slides  through 
its  supports.  Thus  the  contact  is  made  by  turning  the  screw 
until  the  contact-level  is  horizontal.  The  compensating-rod 
is  composed  of  several  concentric  tubes,  alternately  of  brass 
and  iron,  arranged  one  within  the  other,  and  fastened  at  oppo- 
site ends  alternately.  Thus,  when  a  contact  has  been  made  by 
the  fine-motion  screw,  changes  of  temperature  will  not  produce 
derangement,  as  would  be  the  case  if  this  rod  were  not  com- 
pensating. The  arrangement  permits  the  observer  conveniently 
to  work  the  fine-motion  screw,  and  to  observe  its  action  on  the 
contact-level. 

The  apparatus  thus  described  is  enclosed  in  a  double  tin  tubu- 
lar case,  diaphragms  being  adapted  for  supporting  and  strength- 
ening the  whole.  The  air-chamber  between  the  two  cases,  one 


BASE-MEASUREMENTS.  57 

and  a  half  inches  apart,  is  a  great  check  on  heat-variations. 
Three  side-openings,  with  tin  and  glass  doors  in  each  tube, 
permit  observations  of  the  parts  and  of  inserted  thermometers. 
The  ends  are  closed,  only  the  sliding-rod  ends  projecting  at 
each  extremity,  exposing  the  agates.  Brass  guard-tubes  pro- 
tect these,  and  for  transportation  tin  conical  caps  are  screwed  on 
the  tube-ends.  The  fine-motion  screw,  the  sector-tangent  screw, 
and  the  contact-lever-clamp  screw  project  beyond  the  case. 
The  tube  is  painted  white,  which,  with  the  air-chamber  and 
thorough  compensation,  effectually  obviates  all  need  of  a  screen 
from  the  sunshine,  which  has  usually  been  deemed  requisite. 

The  tube  rests  on  a  fore  trestle  and  rear  trestle,  which  are 
alike,  except  in  the  heads.  Each  trestle  has  three  legs,  com- 
posed of  one  iron  cylinder  moving  in  another  by  means  of  a 
rack,  pinion,  and  crank,  so  as  to  raise  or  sink  the  head-plate. 
The  levelling  and  finer  adjustment  are  by  means  of  a  foot- 
screw  in  each  leg,  by  -working  which  a  circular  level  on  the 
connecting-frame  is  adjusted.  A  large  axis-screw,  resting  on 
the  connecting-frame,  and  rising  into  a  tubular  nut,  is  turned 
by  bevelled  pinions  worked  by  a  crank,  and  thus  raises  or  lowers 
this  tubular  nut  and  the  cap-piece  which  it  supports  at  top. 
The  axis-screw,  the  leg-racks,  and  the  foot-screws  give  three 
vertical  movements  in  the  trestle,  by  which  its  capacity  for 
slope-measurements  is  much  amplified. 

In  the  cap  of  the  rear  trestle,  a  lateral  and  a  longitudinal 
motion  are  provided  for,  by  means  of  two  tablets  arranged  to 
slide,  the  upper  one  longitudinally  on  the  lower  one,  and  the 
lower  laterally  on  the  head-plate  of  the  axis-screw  tube.  Long 
adjusting  screw-handles  extend  to  the  observer's  stand  from 
these  two  plates  and  from  the  axis-screw,  enabling  him  to  raise 
or  lower,  to  slide  forward  or  back,  to  the  right  or  the  left,  the 
rear  end  of  the  tube.  The  fore  trestle  is  similar,  except  that 
its  head  is  only  arranged  for  a  lateral  movement,  and  a  second 
observer  makes  its  adjustments  by  a  simple  crank. 

Four  men  can  carry  a  tube,  by  levers  passed  through  staples 


58  GEODETIC  OPERATIONS. 

in  blocks  strapped  under  the  tubes.  The  principal  observer 
and  an  assistant  make  the  contacts  and  rectifications,  the  first 
assistant  directs  the  forward  tube,  and  another  preserves  the 
alignment  with  a  theodolite.  A  careful  recorder  notes  down 
the  observations,  and  an  intelligent  aid  places  the  trestles  and 
foot-plates. 

This  scale  referred  to,  known  as  Borda's  scale,  was  introduced 
in  Bessel's  system,  the  only  difference  being  that  he  used  iron 
and  zinc  in  the  place  of  copper  and  platinum, -and  measured 
the  interval  with  a  glass  wedge.  In  this  the  iron  is  the  longer, 
and  supports  on  its  upper  surface  the  zinc. 

The  zinc  terminates  at  its  free  end  in  a  horizontal  knife-edge, 
and  the  iron  bar  very  near  this  has  attached  to  itself  a  piece  of 
iron  with  a  vertical  knife-edge  on  each  side  in  the  direction  of 
the  length  of  the  bar.  The  distance  between  the  end  of  the 
zinc  and  this  fixed  point,  changing  with  the  varying  tempera- 
ture, is  measured  by  means  of  a  glass  wedge,  whose  thickness 
varies  from  0.07  of  an  inch  to  0.17  of  an  inch,  with  120  divis- 


FlG.    7. 

ions  engraved  on  its  face,  the  distance  between  its  lines  being 
0.03  of  an  inch.  The  other  vertical  knife-edge,  projecting 
slightly  beyond  the  end  of  the  bar,  is  brought,  in  measuring, 
very  near  the  horizontal  knife-edge  in  which  the  opposite  end 
of  the  bar  terminates,  and  the  intervening  distance  measured 
with  the  same  glass  wedge.  If  the  wedge  in  this  case  be  care- 
fully read  and  its  thickness  at  each  division  accurately  known, 
this  method  eliminates  some  of  the  uncertainties  in  the  method 
of  contact.  A  pair  of  Bessel  bars,  slightly  modified,  is  now  in 
use  in  the  Prussian  Landes-triangulation. 


BASE-MEASUREMENTS.  59 

The  annexed  cut  shows  the  arrangement  of  the  knife-edges 
in  the  two  ends  of  the  Bessel  bars. 

The  apparatus  devised  by  Colby  consists  of  a  bar  of  brass 
and  one  of  iron,  fastened  at  their  centres,  but  free  to  move  the 
rest  of  their  lengths.  Each  end  of  one  of  the  bars  is  a  fulcrum 
of  a  transverse  lever  attached  to  the  same  end  of  the  other  bar, 
the  lever  arms  being  proportional  to  the  rates  of  expansion  of 
the  bars.  In  this  way  the  microscopic  dots  on  the  free  ends  of 
the  levers  are  theoretically  at  the  same  distance  apart  for  all 
temperatures.  As  the  terminal  points  were  the  dots  on  the 
lever  arms,  contact  could  not  be  made  in  measuring,  so  the  in- 
terval between  two  bars  was  determined  by  a  pair  of  fixed  micro- 
scopes at  a  known  distance  apart. 

In  all  forms  of  compensating-bars,  the  components  having 
different  rates  of  heating  and  cooling,  their  cross-sections  should 
be  inversely  proportional  to  their  specific  heats,  and  should  be 
so  varnished  as  to  secure  equal  radiation  and  absorption  of 
heat.  Struve's  apparatus  consists  of-  four  bars  of  wrought  iron 
wrapped  in  many  folds  of  cloth  and  raw  cotton. 

Contact  is  made  by  one  end  of  a  bar  abutting  against  the 
the  lower  arm  of  a  lever  attached  to  the  other,  while  the  upper 
arm  passes  over  a  graduated  arc  on  which  a  zero-point  indi- 
cates the  position  of  the  lever  for  normal  lengths.  The  tem- 
perature is  ascertained  from  two  thermometers  whose  bulbs  lie 
within  the  bar. 

From  these  descriptions  it  can  be  seen  that  the  Bache  appa- 
ratus was  a  combination  of  principles  separately  used  before. 
It  had  Borda's  scale,  Colby's  compensation-arm,  and  Struve's 
contact-lever;  with  this  difference:  the  lever,  instead  of  sweep- 
ing over  a  graduated  arc,  acted  upon  a  pivoted  level.  The 
form  used  by  Porro  in  Algiers  consisted  of  a  single  pair  of  bars 
attached  at  their  common  centre  and  free  to  expand  in  both 
directions.  Each  end  of  one  of  the  bars  carried  a  zero-point, 
while  the  corresponding  end  of  the  other  had  a  graduated  scale 
like  Borda's.  In  measuring,  a  micrometer  microscope  is  placed 


60  GEODETIC  OPERATIONS. 

on  a  strong  tripod,  with  an  adjustable  head  immediately  over 
the  initial  point.  The  apparatus  is  then  placed  in  position  on 
another  pair  of  trestles,  completely  free  from  the  microscope- 
stands,  and  moved  by  slow-motion  screws  until  it  is  in  line 
and  the  zero-point  in  the  axis  of  the  microscope.  The  scale  is 
then  read  by  means  of  the  micrometer ;  at  the  same  time  another 
similar  microscope  is  being  adjusted  over  the  forward  end  and 
read.  The  bar  is  then  carried  forward,  placed  in  position  so 
that  its  rear  end  is  under  the  second  microscope,  and  the  for- 
ward end  ready  for  a  third  microscope  previously  aligned. 

And  so  the  work  progresses  until  a  stop  is  to  be  made  ;  then 
the  bar  is  removed  and  a  point  established  under  the  forward 
end  of  the  bar.  Every  precaution  is  taken  to  estimate  flexure 
and  to  avoid  uncertainties  of  collimation  and  unstable  micro- 
scopes. In  Ibafiez's  apparatus  the  component  bars  are  copper 
and  platinum,  mounted  upon  a  double  T-iron  truss.  Flexure 
is  determined  by  resting  a  long  level  on  the  bars  at  several 
points  at  equal  distances  apart.  It  differs  from  the  preceding 
in  having  the  bars  exposed. 

The  Baumann  apparatus,  recently  constructed  for  the  Prus- 
sian Geodetic  Institute,  has  platinum  and  iridium  bars  resting 
on  an  iron  truss,  with  its  entire  length  open  to  the  free  circula- 
tion of  the  air.  Inclination  is  determined  by  a  level  of  preci- 
sion and  flexure  by  a  movable  level. 

There  are  six  microscope  stands,  the  same  number  of  trestles 
for  the  bars,  and  thirty  sets  of  heavy  iron  foot-plates. 

The  latter  are  put  in  position,  and  remain  half  a  day  before 
being  used.  For  each  microscope-stand  there  are  two  tele- 
scopes— one  for  aligning  and  one  for  reading  the  scales.  Six 
skilled  observers  and  about  thirty  laborers  are  needed  in  meas- 
uring. Only  one  base  has  been  measured  with  this  apparatus 
up  to  the  present  time — that  of  Berlin  in  1884 — but  the  results 
are  not  yet  known. 

The  Repsold  differs  from  the  Baumann  apparatus  only  in  a 
few  points,  the  chief  being:  the  component  bars  are  steel  and 


BA  SE-MEA  S  UREMENTS. 


6l 


zinc,  and  the  two  are  suspended  in  a  steel  tube  which  is 
wrapped  in  thick  felt.  The  small  probable  errors  deduced  by 
Ibafiez  and  the  officers  of  the  Lake  Survey  in  the  results  ob- 
tained with  the  metallic-thermometer  principle  appear  to  com- 
mand its  continuance  in  the  construction  of  base-apparatuses. 
But  in  the  Yolo  base  authenticated  temperature  changes  were 
not  always  accompanied  by  corresponding  indications  of  the 
Borda  scale.  In  short,  the  behavior  of  the  zinc  component  was 
so  unsatisfactory  that  a  new  apparatus  for  the  Coast  and  Geo- 
detic Survey  is  under  consideration,  in  which  the  scale-readings 
will  be  omitted,  and  either  a  partly  compensated  pair  of  bars 
or  a  single  carefully  protected  bar  of  steel  adopted  instead, 
with  daily  comparisons  with  a  field  standard.  For  additional 
information  on  the  various  forms  of  base-apparatuses  the  au- 
thorities cited  at  the  end  of  this  chapter  may  be  consulted. 

It  is  interesting  to  note  the  results  of  various  measurements 
under  different  auspices  with  the  same  or  different  forms  of 
apparatuses.  The  following  list  gives  the  most  important: 


Name  of  base. 

Measured  by.                  Apparatus. 

.Length.          I 

'rob.  error. 

Dauphin  Island.. 
Bodies  Island  
Edisto  Island  .... 
Key  Biscayne  
Cape  Sable  
Eppinsr  Plains  — 
Peach  Ridge  
Fire  Ishnd 

U.S.  C.^and  G.  S  

Bache-W  

Hassler.... 
Bessel  

Colby  '  '.". 
Bache-W  

Repsold  .'.'.'.'.'.'.'.'. 
Slide-contact  .  .  . 

6.66   mi 
6-75       ' 
6.66 
3-6 
4 
5-4 

d 

5-5 
2300        m 
2480 
8912.5     fe 

f:I  - 

4.6 
3-8 

17486^51    me 
2400.07 
2440.29 
3318.55 
4061.34 
859-44 
2016.56910 

es. 

[ 

res. 
es. 

res. 
;es. 

410000 

425500 
418600 
454400 
409600 
551600 
561880 
483980 
22800 
16949 

22222 
667000 

833r° 
530000 
2089000 
1148600 

t$L 
6000000 
3500000 
2090000 
3700000 
715000 
1333065 
903784 
1577945 

Kent  Island  
Beverloo  
Ostend  
Cape  Comorin.... 
Keweenaw  
Minnesota  
Chicago  

Nerenberg  

Eng.  Trig.  S   ... 
U.S.  LakeS  

Wingate  

U.  S.  Geol.  S  

Yolo  

U.  S.  C.  and  G.  S  
Ibafiez  and  Hirsch  
Hirsch  
Haffnerand  Overgaard.. 
Kalmer  and  Lehrl  
Schwerd   
Italian  Government  

Schott  
Ibafiez  

Swedish  Acad'y.. 
Austrian  
Schwerd  
Bessel  

Aarberg  
Weinfelden  
Joederen  
Ilidze  
Speyer  
Fosjsria  

Naples  
Axevalla  

Stecksen  ... 

Wrede  

340.224 
I357-033 

GEODETIC  OPERATIONS. 


Perhaps  a  better  idea  can  be  obtained  of  the  accuracy  of 
base-measurements  when  we  give  a  comparison  of  the  measured 
length  of  a  line  with  its  length  as  computed  from  another  base. 
A  few  such  comparisons  are  here  given : 

Epping  measured 871 5-942  metres. 

Computed  from  Massachusetts  base 871 5.865       " 

"     Fire  Island  base 8715.900      " 

Massachusetts  base  measured 17326.376       " 

Computed  from  Epping  base 17326.528       " 

"      Fire  Island 17326.445       " 

Combining  the  errors  of  preliminary  measurements  with  the 
computed  error  in  the  triangulation,  the  appended  results  are 
obtained : 

Probable  error  in  junction-line.  Due  to  base.    To  triangulat'n  Both. 

From  Epping  base 0.17  metre.     0.76  metre.     0.78  metre 

"      Massachusetts  base 0.20       "        0.32       "         0.37       " 

"     Fire  Island  base  0.39       "        0.66       "        0.77       " 


Considering  the  distance  apart  of  these  bases,  it  is  safe  to  say 
that  if  the  errors  are  constant  the  maximum  error  in  the  length 
of  any  line  of  the  triangulation  is  not  more  than  0.22  of  an 
inch  to  the  statute  mile.  The  above  are  the  results  of  measure- 
ments by  the  Bache-Wurdeman  apparatus,  angles  measured 
with  a  thirty-inch  repeating-theodolite,  and  the  triangulation 
computed  by  Mr.  Schott.  Simply  with  the  purpose  of  com- 
paring  the  results  obtained  by  different  apparatus,  I  make  an 
extract  from  the  report  of  the  U.  S.  Lake  Survey : 

Chicago  base  measured log.  in  feet  4.39 17929 

"     computed  from  Fond  du  Lac  "          "     4.3918010 

Difference  =  0.14  metre. 
Distance  from  Chicago  to  Fond  du  Lac,  150  miles. 


BASE-MEASUREMENTS.  63 

Olney  base  measured log.  in  feet  4.3349231 

"         "     computed  from  Chicago "  "    4.3349231 

Difference  =  0.06  metre. 
Distance  from  Chicago  to  Olney,  200  miles. 


The  Madridejos  base,  measured  by  General  Ibaflez  with  his 
improved  Porro  apparatus,  was  divided  into  five  segments;  the 
central  one  was  about  1.75  miles  long.  This  one  was  meas- 
ured twice,  and  used  as  a  base  in  computing  the  length  of  each 
of  the  other  segments.  The  relation  between  the  measured 
and  computed  values  may  be  seen  in  the  following  table : 


Segment. 

Measured  (metres). 

Computed  (metres). 

Difference  (metres). 

I 
2 
3 

4 

5 

Total  

3077-459 
2216.397 
2766.604 
2723.425 
3879.000 

3077.462 
2216.399 
2766.604 
2723.422 
3879.002 

—  0.003 

—    O.O02 

+  0.003 

—    O.OO2 

14662.885 

14662.889 

—    O.OO4 

The  Wingate  base,  measured  with  a  slide-contact  apparatus, 
was  divided  into  three  segments;  the  middle  one  was  measured 
twice  to  see  if  a  discrepancy  sufficiently  great  to  warrant  a  re- 
measurement  existed.  The  two  results  were  in  sufficient  ac- 
cord to  admit  of  the  acceptance  of  the  entire  measurement  as 
correct.  However,  each  segment  was  used  as  a  base  for  the 
computation  of  the  other  segments. 

The  length  of  the  line  was  : 

With  measured  first  and  computed  2d  and  3d. . .  6724.5309  m. 

"  second  "  1st  and  3d.  ..  6723.7132  " 

"  third  "  ist  and  2d. ..  6723.8248  " 

Measured  value  of  the  whole  line 6724.0844  " 


64  GEODETIC  OPERATIONS. 

Giving  these  values  equal  weight,  the  length  may  be  written 
6724.0383  ±0.12  metres. 

Colonel  Everest,  with  the  Colby  apparatus,  measured  in  India 
three  bases,  and  joined  them  in  the  scheme  of  triangulation, 
measuring  the  angles  with  a  thirty-six-inch  theodolite. 

Dehra  Dun.  Damargida. 

Measured  length  in  feet 39l83-87  41578-54 

Computed     "        "         39183-27  41578.18 

Only  instructions  of  the  most  general  kind  can  be  given  for 
the  mechanical  part  of  measuring.  The  details  vary  with  each 
form  of  apparatus.  The  location  of  the  base  is  a  matter  of 
prime  importance,  and  must  be  considered  in  connection  with 
the  purpose  for  which  the  base  is  needed.  If  for  verification, 
it  should  be  suitably  situated  for  connection  with  the  chain  of 
triangles  it  is  intended  to  check. 

If  it  is  intended  to  serve  as  an  initial  base,  a  favorable  con- 
dition for  immediate  expansion  should  be  sought.  As  the 
base  will  usually  be  from  three  to  seven  miles  long,  the  points 
suitable  for  the  first  triangle-stations  should  be  somewhat 
farther  than  that  apart,  permitting  a  gradual  increase  in  the 
lengths  of  the  sides.  The  best  initial  figure  is  undoubtedly  a 
quadrilateral  of  which  the  base  is  a  diagonal,  giving  an  expan- 
sion from  either  side,  or  from  the  other  diagonal. 

If  this  be  impracticable,  the  base  must  be  a  side  of  a  com- 
plete figure.  Of  course  the  termini  must  be  intervisible,  and 
at  the  same  time  visible  from  every  point  of  the  line.  If  the 
ground  is  irregular,  having  slopes  exceeding  three  degrees  in 
inclination,  it  must  be  graded  to  within  that  limit,  with  a 
width  of  about  twelve  feet.  The  method  of  alignment  varies 
with  the  views  of  the  person  in  charge. 

A  good  plan  is  to  select  a  point  approximately  at  the  mid- 
dle of  the  line.  Place  a  theodolite  there,  and  direct  the  tele- 


BASE-MEASUREMENTS.  65 

scope  to  the  temporary  signal  at  one  end  and  read  the  angle 
to  the  other  end  ;  if  it  differs  from  180°,  move  the  instrument 
in  the  proper  direction  until  the  angle  is  just  180°.  Assistants 
are  then  sent  towards  each  end,  and,  from  signals  from  the  per- 
son at  the  instrument,  secure  points  in  line :  these  should  be 
placed  about  a  quarter  of  a  mile  apart.  Considerable  experi- 
ence has  shown  that  the  best  form  of  aligning  signal  is  a  piece 
of  timber  of  suitable  size,  2x4  inches  or  4  inches  square,  driven 
in  the  ground  and  sawed  off  a  few  inches  above  the  surface. 
In  the  top  of  this,  bore  a  hole  at  the  central  point  for  the 
insertion  of  an  iron  pin,  twice  as  long  as  the  hole  is  deep. 
Take  a  corresponding  piece  of  timber  six  or  eight  feet  long  and 
make  a  similar  hole  in  its  end.  It  can  then  be  adjusted  to  the 
stake  in  the  ground,  and  made  stable  by  two  braces,  after  be- 
ing made  perpendicular  by  means  of  a  plumb  line  or  a  small 
theodolite.  The  advantage  of  this  form  of  signal  is  that  it  can 
be  removed  when  the  measuring  reaches  this  point,  and  be  re- 
placed for  a  future  measurement  without  going  to  the  trouble 
of  making  a  second  alignment.  A  plan  of  aligning  differing 
from  this  is  to  have  the  instrument  carefully  adjusted  and 
placed  three  or  four  hundred  yards  from  the  end.  Direct  the 
telescope  to  the  temporary  signal  at  that  point,  turn  it  in  its 
Y's,  or  180°  in  azimuth,  and  fix  a  point  directly  in  line.  Then 
place  the  instrument  over  the  point  so  selected  and  locate 
another  point  in  advance,  and  so  on  till  the  opposite  end  is 
reached.  This  will  only  be  possible  when  one  terminus  has 
been  decided  upon  and  the  general  direction  of  the  line.  Each 
terminus  of  the  base  is  marked  by  a  heavy  pier  of  masonry  of 
secure  foundation  with  upper  surface  eighteen  inches  or  two 
feet  below  the  surface  of  the  ground.  In  the  centre  of  the  large 
stone  forming  a  part  of  the  top  of  the  pier  a  hole  is  drilled ;  in 
this,  with  its  upper  face  even  with  the  top  of  the  stone,  is 
placed,  and  secured  by  having  poured  around  it  molten  lead, 
a  copper  bolt  or  a  piece  of  platinum  wire. 
5 


66  GEODETIC  OPERATIONS. 

On  the  upper  end  of  this  bolt  or  wire  a  needle-hole  may  be 
drilled,  or  a  pair  of  microscopic  lines  drawn,  whose  intersection 
marks  the  end  of  the  base.  Immediately  above  this  should 
be  placed  a  surface-mark  to  which  the  position  of  the  theodo- 
lite can  be  referred  in  the  triangulation  ;  also  a  set  of  witnesses 
consisting  of  four  stones  projecting  above  ground,  so  placed 
that  the  diagonals  intersect  above  the  under-ground  mark. 

When  both  ends  are  marked  in  this  way  before  measuring, 
the  distance  from  the  end  of  the  last  bar  to  the  terminal 
mark,  already  fixed,  is  measured  on  a  steel  scale  horizontally 
placed. 

The  only  advantage  possessed  by  this  method  is,  that  both 
monuments  have  an  opportunity  to  settle  before  the  distance 
between  them  is  determined.  It  is  believed,  however,  that 
greater  inaccuracies  will  result  from  the  uncertainty  in  this 
scale  and  its  use  than  from  the  irregular  settling  of  the  pier 
placed  after  the  measurement  is  finished.  Before  beginning  the 
accurate  measurement  it  is  advisable  to  make  a  preliminary 
measurement  with  a  steel  tape  or  wire,  marking  every  hun- 
dred lengths  of  the  apparatus  to  be  used.  This  will  serve  as  a 
check  upon  the  record  as  the  final  work  advances ;  and  if  the 
line  is  to  be  divided  into  segments  it  will  show  where  the  in- 
termediate monuments  are  to  be  erected.  When  these  inter- 
mediate stations  are  occupied  the  angle  between  the  ends  and 
the  other  points  should  be  measured  with  great  care,  so  that, 
if  the  line  be  found  to  be  a  broken  one,  the  exact  distance  be- 
tween the  termini  in  a  straight  line  can  be  computed.  If  the 
required  distance  cannot  be  obtained  without  crossing  a  ra- 
vine or  marsh,  the  feasible  parts  can  be  measured,  and  the  other 
portion  computed  by  triangulation. 

The  form  of  record  will  of  course  vary  with  the  kind  of  ap- 
paratus used,  but  too  much  care  cannot  be  taken  in  keeping 
the  record.  The  principal  data  needed  in  the  reduction  may 
be  stated  as  follow : 


BASE-MEASUREMENTS.  67 

1.  The  time — showing  the  time  at  which  each  bar  was  placed 
in  position  in  order  to  form  some  idea  of  the  average  speed  at- 
tained in  the  work. 

2.  The.  whole   number    of   the   bar.      When  a   preliminary 
measurement  has  been  made  as  suggested,  the  hundredth  bar 
should  end  near  the  stake  previously  driven  ;  if  not,  a  remeas- 
urement  must  be  made  from  the  last  authentic  point.     This 
should   be  at  the  end  of  the  even-hundred  bar,  and  perhaps 
more    frequently,  especially  if  the  day  should  be  windy,  en- 
dangering the  stability  of  the  bars,  or  if  the  ground  should  be 
boggy  or  springy.     The  simple  method  for  placing  this  point 
is  to  set  a  transit  or  theodolite  at  right  angles  to  the  line  and 
at  a  distance  of  twenty-five  or  thirty  feet  from  it.     After  level- 
ling, fix  the  cross-wires  of  the  instrument  upon  the  end  of  the 
bar;    then,  pointing  the  telescope  to  the  ground,  direct  the 
driving  of  a  stake  in  a  line  with  this  and  with  the   aligning 
telescope.      The   height   of  the  telescope  should  be  half  the 
height  of  the  bar,  so  that  the  focus  need  not  be  changed. 

Then  in  the  top  of  this  stake  a  copper  tack  is  driven,  and  on 
its  upper  face  are  drawn  two  lines  coinciding  with  the  vertical 
threads  of  the  two  instruments.  If  they  are  in  good  adjust- 
ment the  intersection  of  these  lines  will  mark  the  end  of  the 
bar.  A  record  must  always  be  made  when  a  stub  is  thus 
placed.  It  is  also  advisable  to  place  a  stub  under  the  instru- 
ment used  for  this  horizontal  cut-off,  so  that  if  it  should  be 
necessary  to  begin  work  at  this  point  the  instrument  would 
occupy  the  same  position  that  it  occupied  before,  eliminating 
by  this  means  the  error  that  would  arise  from  not  having  the 
transit  at  right  angles  to  the  line. 

Probably  a  more  accurate  method  is  to  have  a  metal  frame 
one  inch  wide  and  two  inches  long  with  screw  holes  admitting 
of  attachment  to  a  stake.  This  frame  has  sliding  inside  of  it 
another  that  can  be  moved  by  a  milled-head  screw,  with  a  set 
screw  to  hold  it  in  place.  On  the  upper  surface  of  this  frame 


68  GEODETIC  OPERATIONS. 

is  a  small  dot  or  hole.  When  the  approximate  position  of  the 
end  is  determined  by  a  plummet,  a  stake  is  driven  in  the  ground 
until  only  an  inch  or  so  remains  above  the  surface :  to  this  is 
attached  the  outer  frame  ;  then,  with  the  theodolite  previously 
set  upon  the  end  of  the  measuring-bar,  direct  the  movement 
of  the  inner  frame  until  the  hole  or  dot  is  bisected  by  the 
cross-wires,  when  the  frame  is  clamped  in  place  and  verified. 
When  microscopes  are  used,  the  dot  can  be  brought  under  the 
micrometer-wire  that  marked  the  position  of  the  zero-point  on 
the  bar. 

3.  The  designation  of  the  bar  as  A,  B,  or  I,  2,  etc.,  so  that 
it  may  be  known  how  many  times  each  bar  was  used.      Since 
the  two  are  never  of  the  same  length,  the  distance  obtained 
by  each  bar  must  be  separately  computed  and  the  two  values 
added  to  get  the  entire  length  of  the  line. 

4.  Inclination.    When  going  up-hill  the  inclination  is  recorded 
plus,  and  minus  when  going  down.     However,  as  the  correc- 
tion for  inclination  is  always  subtracted,  the  sign  is  of  small 
consequence. 

5.  Columns  for  the  sector-error  and  the  corrected  values  for 
the   inclination.     Before   beginning  work  each  day  the  rods 
should  be  placed  on  their  tripods  and  be  made  perfectly  hori- 
zontal by  raising  one  of  them.     To  determine  this,  set  up  a 
carefully  adjusted  theodolite  at  such  a  distance  that  both  ends 
of  the  bar  can  be  seen.     Set  the  thread  on  one  end  of  the  bar, 
revolve  the  instrument  in  azimuth,  and  see  if  the  thread  be 
on  the  other  end :  when  such  is  the  case,  bring  the  bubble  of 
the  sector   in  the  middle  of  the  tube  and  see  what  the  scale- 
reading  is ;  if  zero,  then  there  is  no  error.     This  test  should  be 
applied  at  the  beginning  and  close  of  each  day's  work,  and  the 
average  error  added  to  or  subtracted  from  the  reading  of  in- 
clination for  that  day.     With  secondary  apparatus  this  is  un- 
necessary, as  the  positive  and  negative  readings  will  be  about 
equal,  so  that  the  number  of  readings  that  are  recorded  too 


BASE-MEASUREMENTS.  69 

great  will  be  corrected  by  those  that  are  too  small  by  the  same 
quantity. 

6.  Temperature.     The  thermometers  should  be  read  about 
every  ten  bars,  and  in  the  Borda  rods  the  scales  more  fre- 
quently.    When  the  temperature  gets  above  90°  Fahr.,  it  is 
advisable  to  stop  work,  especially  if  the  bars  are  not  compen- 
sated, as  the  adopted  coefficients  of  expansion  at  that  tem- 
perature are  unreliable. 

The  Repsold  apparatus,  as  used  on  the  Lake  Survey,  and 
the  Davidson,  with  which  the  Yolo  base  was  measured,  were 
protected  during  measuring  by  a  canopy  made  of  sail-cloth 
mounted  on  Wheels,  so  as  to  move  along  as  the  work  advanced. 

In  all  kinds  of  apparatus  it  is  advisable  to  measure  when  the 
bars  indicate  a  rising  temperature,  and  also  during  the  time 
required  for  them  to  fall  through  the  same  amount. 

7.  A  column  for  corrections  for  inclination,  computed  from 
a  formula  to  be  given. 

8.  A  column  for  remarks,  explaining  delays,. stoppages,  the 
placing  of  stubs,  etc. 

GENERAL   PRECAUTIONS  TO   BE  TAKEN  WHILE   MEASURING. 

The  rear  end  of  the  bar  must  be  directly  over  the  marking 
on  the  initial  monument. 

The  inclination  must  never  be  so  great  as  to  endanger  a 
slipping  of  the  bars  forward  or  backward. 

The  trestles  should  be  so  firmly  set  that  there  can  be  no  un- 
equal settling  after  the  bar  has  been  placed  on  them. 

A  bar  should  not  be  allowed  to  remain  more  than  a  minute 
in  the  trestles,  lest  its  weight  should  change  their  position. 

When  a  stoppage  is  made  to  allow  the  aligning-instrument 
to  advance,  a  transit  should  be  set  up,  as  already  described, 
and  its  cross-wires  firmly  clamped  on  the  end  of  the  bar;  then, 
before  resuming  work,  the  position  of  the  bar  can  be  restored 
if  from  any  cause  it  has  changed. 

When  the  end  has  been  transferred  to  a  temporary  mark,  as 


JQ  GEODETIC  OPERATIONS. 

when  a  stop  is  made  for  night  or  dinner,  in  resuming  work  it 
is  best  to  place  the  bar  that  the  work  closed  with  in  the  same 
position  it  had  before  stopping ;  then  the  new  day's  work  goes 
on  as  though  there  had  been  no  break.  If  this  plan  is  not 
adopted,  either  in  the  transferrence  to  the  ground  or  from  it, 
the  end  sighted  will  be  more  than  the  standard  length  from 
the  other  end,  being  held  out  by  the  spiral  spring  that  keeps 
the  agate  beyond  its  proper  distance,  rendering  it  necessary  to 
record  an  index-error  for  every  transferrence ;  whereas  in  the 
plan  suggested  there  can  be  no  danger  of  omitting  to  record 
this  index-error,  nor  of  recording  an  erroneous  value. 

This  precaution  refers  to  that  species  of  apparatus  which 
consists  of  a  pair  of  bars,  one  abutting  against  the  other,  and 
not  where  only  one  bar  is  used,  as  in  the  Repsold,  Baumann, 
and  others. 

The  alignment  must  be  made  with  precision,  for  all  errors  of 
this  kind  are  of  the  same  character  and  do  not  cancel  one 
another. 

Before  beginning  actual  work  the  party  should  measure  a 
short  distance  several  times,  by  way  of  practice,  until  the  dis- 
agreement between  two  measures  is  made  very  small. 

COMPUTATION    OF    RESULTS. 

In  order  to  know  the  horizontal  distance  between  the  two 
ends  of  the  base  it  is  necessary  to  know  the  number  of  times 
the  measuring-unit  was  used,  and  its  exact  length  each  time 
that  it  was  employed.  To  this  must  be  added  index-errors, 
and  the  amount  by  which  the  last  bar  fell  short  of  the  ter- 
minus. Also,  there  are  to  be  subtracted  the  quantities  that 
were  needed  to  reduce  each  length  to  its  horizontal  projection, 
and  those  negative  errors  that  could  not  be  obviated. 

A  carefully  kept  record  will  show  how  often  the  bars  were 
used  ;  but  to  ascertain  their  length  is  a  more  difficult  problem, 
depending  upon :  (a),  a  knowledge  of  the  exact  length  of  the 
adopted  standard  ;  (&),  a  known  relation  between  the  measuring- 


BASE-MEASUREMENTS.  /I 

bar  and  the  standard  at  a  certain  temperature ;  (c\  a  knowledge 
of  the  temperature  of  the  bars  each  time  used,  and  the  coeffi- 
cients of  expansion. 

The  Committee  Metre  is  the  standard  of  linear  measures 
now  in  use,  and  with  a  certified  copy  of  this,  all  our  units  are 
compared.  This  comparison  can  be  described  only  in  outline. 
We  have  two  firmly  built  pillars  at  a  convenient  distance  apart 
for  the  bars  that  are  to  be  compared.  On  one  is  an  abutting- 
surface,  and  on  the  other  is  a  comparator.  In  general,  this 
comparator  consists  of  a  pin  held  out  by  a  spiral  spring  but 
capable  of  being  withdrawn  by  a  micrometer-screw.  This  pin 
works  a  lever  on  whose  longer  arm  is  a  point  that  is  to  be 
brought  into  coincidence  with  a  fixed  zero-mark.  Between 
these  two  pillars  is  a  carriage  rigidly  constructed  but  com- 
pletely isolated  from  them.  On  this  carriage  are  placed  the 
standard  and  the  bar  that  is  to  be  compared.  The  former  is 
placed  between  the  abutting-surface  and  the  micrometer-pin, 
the  screw  is  turned  until  the  zero-marks  coincide,  and  the  turns 
and  division  recorded. 

The  carriage  is  then  moved  along  until  the  bar  is  brought 
into  place  and  the  micrometer  is  again  read.  The  difference 
in  the  readings  will  correspond  to  the  difference  in  lengths  in 
terms  of  micrometer  turns  and  divisions — the  value  of  a  turn 
and  a  division  being  found  by  measuring  with  the  screw  the 
length  of  a  standard  centimetre.  In  very  accurate  comparisons 
the  bars  are  immersed  in  glycerine,  which  can  be  readily  kept  at 
the  same  temperature  for  a  long  time. 

The  temperature  is  ascertained  from  three  thermometers — 
one  at  each  end,  and  one  at  the  middle.  Also,  to  eliminate 
accidental  errors,  a  number  of  readings  are  made  with  the  bars 
reversed,  turned  over,  taken  in  different  order,  and  at  different 
temperatures.  The  average  difference  will  be  the  difference 
in  the  lengths  of  the  standard  and  the  bar  at  the  average  tem- 
perature, supposing  that  the  coefficients  of  expansion  remain 
constant.  Then  knowing  the  temperature  at  which  the  stand- 


72 


GEODETIC  OPERATIONS. 


ard  is  correct  and  its  coefficient  of  expansion,  its  true  length 
can  readily  be  computed  for  this  average  temperature.  To 
this,  add  the  average  difference  just  referred  to  and  we  have  the 
exact  length  of  our  bar  at  this  mean  temperature.  To  illus- 
trate: let  M  be  the  standard,  A  the  bar  under  comparison,  p 
the  difference  in  microns,  which  is  obtained  by  multiplying  the 
turns  and  divisions  of  the  micrometer  by  the  previously  ascer- 
tained value  of  one  turn. 


Temp. 

A  -  M. 

57.58 
52.60 
55-29 
55-16  =  /. 

+   ?'-  5 

+  7-5 
+  9-8 
+  8.27 

Therefore,  A  —  M-\-  8.27/1  at  55°.  16.  Suppose  e  be  the  co- 
efficient of  expansion  for  M,  and  T  the  temperature  at  which 
Mis  correct,  then  we  have  A=M  +Me($$°.\6  —  T)  +  8.27/1. 

To  determine  e  we  must  have  the  pillars  of  the  comparator 
at  a  fixed  distance  apart,  and  then  measure  this  distance  with 
a  bar  at  different  temperatures.  In  order  to  insure  the  bar 
being  at  the  same  temperature,  it  is  best  to  place  it  in  glycer- 
ine previously  heated,  and  leave  it  there  for  half  an  hour.  Let 
D  be  the  difference  between  the  constant  distance  and  the  dis- 
tance as  observed  at  various  temperatures,  /„  the  average,  and 
t  the  observed  temperatures. 


t. 

D. 

,-,„      ,-A, 

•>  F. 
99.08 

441-5 

+  28.13^  = 

178.9 

83.68 

342.9 

12.73^  = 

80.3 

72.08 

268.2 

1.07^  = 

5-6 

57-58 

175-9 

-   13-37'=  - 

86.7 

42-39 

084.5 

—  28.56^  =  — 

178.1 

70.95  =  /, 

262.6  =  Z?0 

BA  SE-MEA  SUREMENTS. 


73 


Forming  the  normal  equations  by  multiplying  each  equation 
by  the  coefficient  of  e  in  that  equation,  and  taking  the  sum  of 
the  resulting  equations,  we  get  1948.92^  —  12,306.4^,  or 
e  =  6.315 /*.  Substituting  this  value  of  e,  we  have  for  A, 
A  :=  J/  + 6.315/^(55°. 16  —  T)  +  8.2;//.  There  is  a  probable 
error  in  this  determination  which  can  be  carried  through  the 
future  computations. 

The  way  in  which  the  temperature-observations  are  utilized 
depends  upon  the  accuracy  desired ;  ordinarily  the  average 
temperature  of  each  bar  in  a  segment  is  employed.  So  that  if 
we  have  n  lengths  of  a  four-metre  bar  with  the  above  coeffi- 
cient of  expansion,  a  length  equal  to  A  at  55°.i6,  and  the  aver- 
age temperature  t  in  that  segment,  we  shall  have  the  distance 
=  n[A  +  4X0.000  006315  (/— 55°.i6)].  When  greater  ac- 
curacy is  required,  the  length  of  each  bar  can  be  computed  in 
the  same  manner,  and  the  aggregate  length  obtained  by  sum- 
mation. 

When  a  Borda  scale  or  metallic  thermometer  is  used,  it  is 
necessary  to  know  how  much  in  thermometric  scale  a  division 
is  equal  to.  The  scale  is  usually  divided  into  millimetres,  and 
read  by  a  vernier  or  microscope  to  o.oi  mm. 


Temp. 

*-/„• 

Scale  =  S. 

JS. 

o  p 

109.41 
94.11 

+  31-79 
+  16.49 

8.60 

8.17 

+  0.92 
+  0.49 

79-21 

+     1-59 

7-74 

-f  0.06 

61.16 

-  16.46 

7.16 

—  0.52 

44.22 

-  33-40 

6.72 

—  0.96 

77-62  =  /» 

7.68  =  S, 

By  letting;?  be  the  quantity  representing  the  differential  ex- 
pansion of  the  component  bars,  and  as  it  varies  with  the  tem- 
perature, we  may  take  the  values  of  t  —  /„  as  the  coefficients 
of  x  and  solve  by  least  squares.  The  normal  equation  will 
give  2671.54*-  =  78.05^,  or  x  —  0.02922^  =  29.22^. 


74  GEODETIC  OPERATIONS. 

That  is,  a  change  of  one  degree  Fahr.  is  represented  by 
0.029  division,  or  the  smallest  value  that  can  be  estimated 
on  the  vernier,  o.oid  •=  i°F.;  consequently  the  scale-readings 
can  be  readily  converted  into  degrees  of  temperature  and  the 
reduction  for  length  made  as  in  the  preceding  case,  or  the 
change  in  length  may  be  found  directly  in  terms  of  scale-read- 
ings. If  we  have  a  four-metre  bar  with  the  coefficient  of  ex- 

pansion just  found,  o.o\d~  —  ^—  ^  —  J*  =  8.64^. 

Then  if  S0  be  the  scale-reading  at  wrhich  M  is  a  standard, 
and  S  any  other  reading  during  the  measurement  or  the  aver- 
age, A  =  M-\-  8.64/^(5  —  £„),  and  the  entire  line 

=  n[M+  8.64X5  -5.)]. 

Correction  for  inclination  :  if  R  represent  the  length  of  a 
bar,  h  its  horizontal  projection,  and  0  the  angle  of  inclination, 
it  is  apparent  that  h  =  R.  cos  0,  then  d  the  correction  —  R  —  h 
=  R  -  R  .  cos  0  =  R(i  —  cos  0)  =  2R.  sin2  $6.  As  6  is  small 
sin2  %0  =  -J  sin2  6  (nearly),  so  we  may  write 

Rsin'O       sin"  i'  sin2  i' 

d  =  -  -  -  =  -—R6*',        log  —  —  =  2.626422. 


Having  determined  by  comparison  the  average  length  of  the 
bars,  a  table  should  be  computed  for  each,  giving  the  values 
for  d  for  each  fractional  part  to  which  the  sector  can  be  read, 
and  within  the  limits  observed.  Then  from  this  table  correc- 
tions for  inclination  can  be  taken  and  inserted  in  the  record- 
book.  If  there  are  any  index-errors,  as  stated  might  occur  in 
the  transferrence  of  the  end  to  the  ground,  they  must  be  added 
to  the  computed  length. 

Probable  error.     This  may  be  derived  — 

I.  By  measuring  the  base  a  number  of  times,  then  deducing 


BASE-MEASUREMENTS.  75 

the  probable  error  in  accordance  with  the  principle  of  least 
squares. 

2.  By  dividing  the  line  into  segments  and  computing  the 
other  segments  from  each  one  as  a  base  by  triangulation. 

3.  By  checking  one  base  from  another  in  the  chain  of  trian- 
gulation, and  determining  the  probable  error  in  the   second 
from  that  of  the  first  and  of  the  measurement  of  the  angles  in 
the  triangulation. 

4.  From  all  known  sources  of  error  in  measurement. 

The  fourth  method  is  the  only  one  that  needs  expansion  at 
this  point.  The  principal  sources  of  error  in  measurement 
are: 

1.  In  determining  the  length  of  the  bar. 

2.  Backward  pressure. 

3.  Error  of  alignment. 

4.  In  transferring  end  to  the  ground. 

5.  In  the  determination  of  inclination. 

6.  Personal  errors  of  the  observers. 

These  are  determined  as  follows :  the  first  is  obtained  from 
repeated  comparisons  with  the  standard,  and  is  made  up  of 
two  parts — uncertainty  in  the  expansion  of  the  bars,  and  acci- 
dental errors  in  comparing.  Of  these  the  former  is  found  from 
the  residuals  in  the  series  of  determinations  of  the  coefficients 
of  expansion.  Calling  this  rf,  we  have  for  the  entire  n  bars 
n.r,'.  Likewise  the  error  from  comparison  is  found  in  a  simi- 
lar manner  from  the  series  of  comparisons,  if  we  designate  this 
rt',  the  entire  error  rt  =  n.rj. 

The  error  of  contact  depends  upon  the  force  with  which  the 
agate  is  held  out  beyond  its  proper  position.  When  a  bar  is 
in  its  right  place,  and  the  next  bar  brought  into  contact  with 
it,  the  pressure  necessary  to  bring  it  to  its  place  forces  the  rear 
bar  backward ;  and  when  the  rear  bar  is  taken  away  the  for- 
ward bar,  being  relieved  of  this  pressure,  moves  back  by  the 
same  amount.  Consequently  the  total  backward  movement  is 


76  GEODETIC  OPERATIONS. 

double  the  effect  of  pressure.  This  must  be  determined  by 
experiment  in  various  positions  of  the  bar.  As  every  bar  ex- 
cept the  first  and  last  are  doubly  affected,  these  each  bein  • 
changed  only  once  by  this  pressure,  the  total  correction  wL. 
be  twice  the  displacement  multiplied  by  one  less  than  the 
number  of  bars.  Usually  this  is  too  small  to  be  considered, 
and  applies  to  those  bars  only  that  are  used  in  pairs — one  bear- 
ing in  contact  against  the  other. 

By  (3)  is  not  meant  the  uncertainty  of  having  the  line  as  a 
whole  straight,  but  in  placing  the  bar  exactly  in  that  line.  The 
aligning  instrument  is  placed  in  front  at  distances  varying  from 
50  to  900  feet,  and  the  alignment  is  effected  by  bringing  the 
agate  of  the  bars  into  coincidence  with  the  vertical  thread  of 
the  telescope ;  or  when  the  bars  are  provided  with  a  vertical 
rod  immediately  over  their  centres,  this  is  sighted  to.  It  is 
apparent  that  the  bisection  of  this  may  not  be  perfect ;  and,  in 
fact,  when  the  light  falls  unequally  upon  the  object  sighted  to, 
the  illuminated  spot  is  bisected,  which  may  be  altogether  to 
one  side  of  the  centre. 

However,  the  error  of  bisection  cannot  be  greater  than  the 
radius  of  the  agate  or  aligning-rod,  and  its  effect  upon  the  true 
length  of  the  line  will  depend  upon  the  distance  to  the  transit. 
The  nearer  the  transit,  the  less  is  the  likelihood  of  making  an 
erroneous  bisection.  By  placing  a  scale  directly  under  the 
agate,  and  having  the  person  at  the  transit  direct  the  mov- 
ing of  the  bar  until  he  considers  it  in  line,  make  a  note  of  the 
scale-reading,  and  after  a  number  of  trials  the  average  variations 
may  be  taken  as  the  error  most  likely  to  be  committed  at 
that  distance.  Suppose  it  was  found  that  the  errors  were  a 
for  the  maximum,  and  b  for  the  minimum  distances,  the  an- 
gular variations  might  be  written  :  a  times  one  second  divided 
by  the  length  of  the  bar,  call  this  m,  and  similarly  for  b,  which 
we  will  call  n.  The  correction  for  this  deviation  will  be  the 
difference  between  the  length  of  the  bar  and  the  vertical  pro- 


BASE-MEASUREMENTS,  77 

jection  for  this  angular  deviation.     As  already  shown,  this  is 

R  .  sin4  ///           R  .  sin2  n 
equal  to ,  and . 

Only  the  first  and  last  few  bars  of  each  segment  will  need 
to  have  this  total  lateral  correction  applied  ;  for  the  remaining 
bars  it  will  be  sufficient  to  take  the  average  of  m  and  n,  in  t^e 
formulae  just  given.  As  the  total  correction  from  this  cause 
will  never  amount  to  a  tenth  of  an  inch,  it  is  usually  omitted, 
and  its  probable  error  is  never  considered. 

The  error  from  the  fourth  source  is  determined  from  experi- 
ment, as  in  the  preceding  case.  Suppose  it  is  0.082  mm.; 
as  there  is  a  double  transfer,  the  entire  error  will  be  0.082 
\/~2  mm.  =  o.i  i  mm.,  and  the  total  for  n  bars  will  be  o.ii 
Vn.  mm.  =  r3. 

The  fifth  source  of  error  is  quite  apparent.  The  sector  that 
shows  the  inclination  usually  reads  to  single  minutes,  some- 
times to  ten  seconds.  As  it  is  impracticable  to  obtain  more 
than  one  reading  for  each  inclination,  there  is  an  uncertainty 
as  to  its  correctness.  This  will  vary  with  the  skill  of  the  ob- 
server and  the  character  of  the  sector  used.  The  probable 
error  of  a  single  determination  should  be  ascertained  as  fol- 
lows :  place  the  bar  firmly  in  its  trestles  and  make  several 
readings  of  the  scale  when  the  bubble  of  the  level  is  in  the 
same  position.  From  a  number  of  such  scale-readings  the 
probable  error  is  deduced  in  the  usual  manner. 

To  determine  the  effect  of  this  error  on  the  computed  cor- 
rections for  horizontal  projections,  the  average  observed  in- 
clination must  be  approximated.  Suppose  this  to  be  2°,  the 
probable  error  of  inclination  30",  and  the  length  of  the  bar  R. 
It  has  already  been  shown  that  the  correction  for  inclination  d 
=  R(i  —  cos  9}.  As  6  in  this  case  is  taken  as  2°,  an  approxi- 
mate value  for  the  change  in  d  by  a  mistake  of  30"  in  Q 
can  be  computed  by  getting  d'  when  6  =  6  ±  30";  d'  = 
R\\  —  cos(ft-|-  30")],  and  the  probable  error  in  any  one  deter- 


78  GEODETIC  OPERATIONS. 

mination  wijl  be  the  difference  between  d  and  d'  or  rt',  rt  = 
e  Vn  where  n  =  the  number  of  bars  and  e  =  d  —  d'. 

To  recapitulate :  those  errors  that  are  known  to  exist  and 
the  direction  of  whose  effect  is  unmistakably  determined  can  be 
applied  in  the  reduction  of  the  length  of  the  base,  while  those 
that  are  merely  probable  must  be  used  simply  in  obtaining  the 
probable  error  of  the  measurement  as  a  whole.  The  value  for 
the  length  of  the  base  must  be  diminished  by  the  amount  of 
backward  pressure,  errors  of  alignment,  and  errors  of  inclina- 
tion ;  but  the  remaining  errors  having  a  double  sign  must  be 
regarded  as  probable ;  if  individually  they  be  represented  by 
*V  ?v  r»  •  •  •  rn,  and  the  total  error  by  R,  we  will  have 


R  = 


As  the  sides  of  the  triangulation  are  at  different  elevations 
and  the  base  and  check-base  not  on  the  same  plane,  it  is  neces- 
sary to  know  their  lengths  at  some  common-datum  plane. 
This  by  common  consent  is  the  half-tide  level  of  the  ocean. 

bg,  height  above  half-tide  =  h  ; 

ae,  the  half-correction  for  reduction  =  -  ; 


ae  :  ed  :  :  ab  :  be  ; 


ae  = 


ed.ab 


be 


2ab 


B 


•2ae  =  c  =  h  . 


bgis  so   small  in   comparison  with  eg 
that  it  may  be  omitted,  and  we  write  ; 


BASE-MEASUREMENTS.  79 

h.B  Jh        V\ 

=  B  -7^ ™  ,  where  R  =  radius  of 


radius  of  curvature  \/? 

curvature  at  the  mean  latitude  of  the  base. 

From  the  corrected  value  for  the  length  of  the  base  c  is  to 
be  subtracted.  If  the  elevation  of  the  base  was  found  by  dif- 
ferent methods,  or  from  different  bench-marks,  an  uncertainty 
may  arise  in  the  value  of  h,  giving  a  probable  error  for  c. 

REFERENCES. 

U.  S.  Coast  and  Geodetic  Survey  Reports  as  follows:  1854, 
pp.  103-108;  '57,  pp.  302-305  ;  '62,  pp.  248-255  ;  '64,  pp.  120- 
144;  '73,  pp.  123-136;  '80,  pp.  341-344 ;  '81,  pp.  357-358;  '82, 
pp.  139-149;  also  pp.  107-138  ;  '83,  pp.  273-288. 

Clarke,  Geodesy,  pp.  146-173. 

Report  of  U.  S.  Lake  Survey,  pp.  48-306. 

Zachariae,  Die  Geodatische  Hauptpunkte,  pp.  79-110. 

Jordan,     Handbuch    der   Vermessungskunde,   vol.    ii.    pp. 

73-113- 

Experiences  Faites  avec  1'Appareil  a  Mesurer  les  Bases. 

Compte  Rendu  des  Operations  de  la  Commission  pour  e"ta- 
lonner  les  Regies  employes  a  la  Mesure  des  Bases  Geod£siques 
Beiges. 

Westphal,  Basisapparate  und  Basismessungen. 

Gradmessung  in  Ostpreussen,  pp.  1-58. 


80  GEODETIC  OPERATIONS. 


CHAPTER  IV. 

FIELD-WORK  OF  THE  TRIANGULATION. 

SUPPOSING  that  a  base  has  been  carefully  measured,  or  the 
distance  between  two  stations  previously  occupied  accurately 
known,  the  next  thing  to  be  done  is  to  lay  out  a  scheme  of  tri- 
angles covering  the  desired  territory.  Their  arrangement  into 
figures  depends  upon  : 

1.  The  special  purpose  of  the  work. 

2.  The  character  of  the  country  over  which  the  system  is 
to  be  extended. 

If  the  object  is  to  measure  arcs  of  a  meridian  or  of  a  par- 
allel, for  the  purpose  of  determining  the  figure  of  the  earth, 
great  care  should  be  exercised  in  selecting  triangles  that  are 
approximately  equilateral ;  for  if  in  the  computation  a  very 
long  side  is  to  be  computed  from  a  short  one,  an  error  in  the 
latter  will  be  greatly  magnified  in  the  former.  If  the  purpose 
is  simply  to  meet  the  wants  of  the  topographer,  the  stations 
should  be  selected  with  special  reference  to  his  needs  and 
without  regard  to  the  character  of  the  figures  thus  formed. 
In  an  open  prairie  where  signals  have  to  be  erected  without 
any  assistance  from  natural  eminences,  their  arrangement  may 
be  made  in  strict  accord  with  theoretical  preference. 

The  plainest  system  of  the  composition  of  triangles  into 
figures  is  a  single  string  of  equilateral  triangles  which  possess 
the  advantages  of  speed  and  economy  of  time  and  labor.  Hex- 
agonal figures  are  preferred  by  some,  but  the  general  prefer- 
ence is  for  quadrilaterals  with  both  pairs  of  diagonal  points  in- 
tervisible.  This  system  covers  great  area  and  insures  the 
greatest  accuracy. 


FIELD-WORK  OF   THE    TRIANGULATION.  8 1 

Equilateral  triangles  will  furnish  nine  conditions. 

Hexagons,  with  one  side  in  common,  twenty-one  conditions. 

Quadrilaterals,  twenty-eight  conditions,  covering  the  same 
area  (approximately). 

Signals. — After  deciding  upon  the  positions  of  the  stations, 
the  next  subject  for  consideration  is  the  kind  of  signals  to  be 
used.  In  short  sights,  the  best  form  is  either  a  pole  just  large 
enough  to  be  seen,  or  a  heliotrope  fixed  on  a  stand  or  a  tripod 
carefully  adjusted  to  the  centre  of  the  station.  As  the  helio- 
tropers  are  usually  persons  with  but  little  experience,  range- 
poles  should  be  previously  set,  enabling  them  to  point  their 
instruments  with  some  degree  of  precision. 

(For  a  description  of  the  heliotrope,  its  adjustments,  and  use, 
see  page  45.) 

Owing  to  the  fact  that  there  are  so  many  days  during  which 
it  is  impossible  to  use  the  heliotrope,  and  also  the  additional 
trouble  that  frequently  when  the  sun  is  shining  the  air  is  so 
disturbed  that  the  object  sighted  is  too  unsteady  to  bisect  with 
any  certainty,  the  effort  is  constantly  being  made  to  devise 
some  form  of  night  signal  to  take  the  place  of  day  signals. 

The  great  obstacle  to  the  successful  solution  of  this  problem 
has  been  the  dimness  or  expense  of  the  lights  that  have  been 
tried,  such  as  oil-lamps,  magnesium,  or  electric  lights.  In  June, 
1879,  Superintendent  Patterson  of  the  U.  S.  Coast  and  Geo- 
detic Survey  directed  Assistant  Boutelle  to  make  an  exhaustive 
series  of  observations  with  the  various  methods  of  night  signals, 
with  a  view  to  determine  the  most  effective  method  to  be  used 
in  triangulation. 

The  special  points  to  be  considered  were  : 

1.  Simplicity  and  cheapness. 

2.  Adaptability  to  the  intelligence  of  the  men  usually  em- 
ployed as  heliotropers. 

3.  Ease  of  transportation  to  heights. 

6 


82  GEODETIC  OPERATIONS. 

4.  Penetration,  with  least  diffraction  and  most  precision  of 
definition. 

5.  The  best  hours  for  observation. 

6.  Lateral  and  vertical  refraction,  and  the  extent  to  which 
the  rays  are  affected  by  the  character  of  the  country  over 
which  they  pass. 

An  accurate  account  of  the  various  experiments  made  by 
Captain  Boutelle  are  given  in  Appendix  8  of  the  C.  and  G.  S. 
Report  for  1880.  I  shall  take  the  liberty  of  quoting  his  con- 
clusions ;  they  are : 

"  The  experience  of  the  past  season  enables  me  to  state  with 
some  precision  the  cost  of  the  magnesium  light,  so  much  supe- 
rior to  every  other  yet  tried. 

"  The  success  in  two  instances  of  burning  the  light  by  a  time- 
table established  that  method  as  perfectly  practicable. 

"  It  reduces  the  time  of  burning  it  to  twenty  minutes  per 
hour,  or  to  eighty  minutes  for  four  hours'  observation.  With 
a  delivery  of  ribbon  of  fifteen  inches  per  minute,  the  cost  will 
be  two  dollars  per  night  for  each  light  used.  The  average 
number  of  primary  stations  observed  upon  at  any  one  station 
is  six,  of  which  three  would  require  the  magnesium  light, 
making  the  expense  six  dollars  per  night.  The  nights  when 
observation  would  be  practicable  and  the  lights  burned  may  be 
taken  as  averaging  three  in  a  week,  or  seven  at  each  station. 

"Apart  from  the  first  cost  of  apparatus,  we  should  therefore 
have  as  the  additional  outlay  for  night  observation  for  a  pri- 
mary triangulation  : 

"  I.  Additional  pay  of  six  heliotropers $3-OO 

"  2.  cost  of  burning  three  magnesium  lights 

every  other  night 3.00 

"3.           "           cost  of  kerosene-oil  for  three  lamps...     0.20 
"4.  "  cost  per  day  for  supplies,  etc 0.80 


"  Whole  additional  daily  cost t $7.00 


FIELD-WORK  OF   THE    TRIANGULATION.  8$ 

"  To  offset  this  additional  party  expense  there  will  be  : 

"  i.  The  shortening  of  time  required  in  occupation  of  each 
station  by  the  addition  of  four  hours  of  observing  each  clear 
day  after  sunset.  The  average  time  of  observation  each  day 
being  two  hours,  this  time  will  be  tripled  on  each  clear  day 
and  night. 

"  2.  Necessity  for  encamping  at  many  stations  may  be  avoid- 
ed, where  now  the  probabilities  of  a  long  detention  and  the 
lack  of  any  decent  quarters  within  a  reasonable  distance  require 
the  transportation  and  use  of  equipage. 

"The  conclusions  to  which  the  experiments  and  results  have 
led  me  may  be  generally  summed  up  as  follows : 

"  i.  That  night  observations  are  a  little  more  accurate  than 
those  by  day,  but  the  difference  is  slight  so  far. 

"  2.  That  the  cost  of  apparatus  is  less  than  that  of  good 
heliotropes. 

"  3.  That  the  apparatus  can  be  manipulated  by  the  same 
class  of  men  as  those  whom  we  employ  as  heliotropers. 

"  4.  That  the  average  time  of  observing  in  clear  weather 
may  be  more  than  doubled  by  observing  at  night,  and  thus 
the  time  of  occupation  of  a  station  proportionally  shortened. 
Hazy  weather,  when  heliotropes  cannot  show,  may  be  utilized 
at  night. 

"  5.  That  reflector-lamps,  or  optical  collimators,  burning  coal- 
oil,  may  be  used  to  advantage  on  lines  of  43.5  miles  and  under. 
But  for  longer  lines  the  magnesium  lights  will  be  best  and 
cheapest,  as  being  the  most  certain. 

"  6.  That  for  the  present  we  should  keep  up  both  classes  of 
observation,  both  by  day  and  night ;  and  that  the  observers  in 
charge  of  the  various  triangulations  should  be  informed  of 
the  progress  already  made,  and  encouraged  to  improve  on  the 
methods  and  materials  thus  far  employed  in  night  observa- 
tions." 

At  this  time  many  of  the  parties  in  charge  of  triangulation- 


84  GEODETIC  OPERATIONS. 

work,  under  the  auspices  of  the  Coast  Survey,  make  night  ob- 
servations. The  wisdom  of  this  plan  is  duly  appreciated  by 
all  who 'have  observed  in  the  Eastern  or  Middle  States. 

It  might  be  safely  said  that  more  time  is  spent  in  waiting  for 
suitable  weather  than  in  reading  the  angles,  and  any  means  for 
diminishing  this  waste  will  be  gladly  adopted,  especially  by 
those  who  have  had  their  patience  taxed  by  having  to  wait  day 
after  day  for  the  haze  to  pass  by. 

For  short  sights  or  for  secondary  triangulation  a  reflecting- 
surface,  such  as  a  tin  cone,  will  be  sufficient.  Still  better  is  a 
contrivance  made  of  tin,  in  the  shape  of  the  children's  toy,  that 
is  made  to  revolve  by  a  current  of  air,  and  fixed  on  an  axis  in 
the  top  of  a  pole  or  tree.  If  it  is  of  the  proper  shape,  in  turn- 
ing it  will  catch  the  sun's  rays  at  the  right  angle  to  send  a  re- 
flection to  the  desired  point,  except  when  the  sun  is  on  the 
opposite  side  from  the  observer.  In  lines  still  shorter  a  simple 
pole,  supported  by  a  tripod,  or  a  straight  tree  will  answer. 
Care  must  be  taken,  however,  to  have  the  pole  or  tree  no  larger 
than  is  necessary  to  render  it  visible,  as  large  bodies  are  diffi- 
cult to  bisect.  A  diameter  of  6  inches  will  subtend  an  angle 
of  one  second  at  a  distance  of  20  miles  ;  for  40  miles,  12.3  inches ; 
and  at  60  miles,  18.5  inches.  Sights  have  been  made  upon 
a  tree  12  inches  in  diameter  at  a  distance  of  55  miles. 

Much  time  can  be  gained  and  accuracy  secured  by  making 
the  observations  at  the  most  favorable  time.  For  instance,  if 
a  pole  is  to  be  sighted,  the  proper  time  is  in  the  morning  when 
looking  towards  the  east,  and  in  the  evening  when  looking 
westward.  If  a  reflecting  object  is  used,  the  opposite  rule  to 
the  above  must  be  followed. 

It  is  frequently  necessary  to  elevate  the  instrument  and  ob- 
server in  order  to  obtain  a  longer  length  of  line,  or  to  overcome 
some  impediment.  Fig.  9  will  give  an  idea  of  the  form  that 
has  been  found  most  convenient.  When  it  is  to  be  constructed 
on  a  hill  or  mountain,  it  will  be  found  advisable  to  cut  the 


FIELD-WORK  OF   THE    TRIANGULATION.  8$ 


FIG.  9. 


86  GEODETIC  OPERATIONS. 

timbers  at  the  bottom,  in  order  to  save  the  transportation  of 
useless  materials. 

In  order  to  secure  the  requisite  stability,  and  to  prevent 
shaking  of  the  instrument  by  the  observers  moving  around,  it 
is  necessary  to  have  a  double  structure — one  for  the  theodolite, 
and  one  to  support  the  platform  for  the  party  observing.  For 
a  low  structure  the  form  used  by  the  Prussian  Geodetic  Insti- 
tute will  be  found  sufficiently  firm.  It  is  a  vertical  piece  of  tim- 
ber to  support  the  instrument,  braced  by  a  tripod,  the  whole  sur- 
rounded by  a  quadrangular  platform.  But  when  a  height  of 
more  than  twenty  feet  is  needed,  the  kind  devised  by  Mr. 
Cutts,  and  improved  by  Captain  Boutelle,  will  be  found  more 
satisfactory. 

I  have  worked  on  several  of  this  pattern,  and  can  vouch  for 
their  rigidity ;  and  when  an  awning  is  attached  to  the  legs  of 
the  scaffold  to  shade  the  tripod,  the  unfortunate  results  of 
"  twist "  from  the  action  of  the  sun's  rays  are  avoided. 

From  a  glance  at  Fig.  9  it  will  be  seen  that  the  signal  con- 
sists of  two  parts — a  tripod  and  a  square  scaffold.  It  is  the 
average  experience  that  a  safe  signal,  strong  enough  to  with- 
stand the  heaviest  winds  we  have,  should  be  built  of  timbers 
6  by  8  inches,  with  diagonal  braces  2  by  2  and  3  by  3.  The 
size  of  the  base  is  a  function  of  the  altitude,  a  good  ratio 
being  one  foot  radius  for  every  eight  feet  of  elevation.  The 
legs  of  the  tripod  should  be  set  three  feet  in  the  ground,  and 
would,  if  continued,  meet  at  a  point  four  feet  above  the  plat- 
form. So  that  for  a  signal  whose  scaffold  is  to  be  eighty  feet 
above  the  station-surface  we  would  have  eighty-seven  feet  for 
the  vertical  height  of  the  tripod,  and  the  radius  of  the  base 
would  be  ^--(-0.67  ft.  =  1 1. 54 ft. 

To  lay  out  the  base,  drive  a  stub  in  the  ground  at  the  cen- 
tral point,  and  with  a  radius  equal  to  that  computed  describe 
a  circle  ;  mark  off  on  this  circumference  points  with  a  chord 
equal  to  the  radius,  and  the  alternate  points  will  be  the  places 
for  the  feet  of  the  tripod.  With  a  level,  or  an  instrument  that 


FIELD-WORK  OF   THE    TRIANGULATION.  8? 

can  be  used  as  a  level,  the  bottom  of  the  holes  for  the  tripod 
can  be  placed  on  the  same  plane,  by  marking  on  a  rod  a  dis- 
tance that  is  equal  to  the  height  of  the  axis  of  the  instrument 
and  three  feet  more,  then  the  holes  are  to  be  dug  until  this 
mark  coincides  with  the  cross-wires  of  the  telescope  when  the 
rod  is  in  each  hole. 

The  tripod,  being  the  highest  and  the  innermost  structure, 
should  be  raised  first.  The  plans  adopted  for  this  differ  with 
different  persons ;  some  frame  two  legs  with  their  bracing,  raise 
them  with  a  derrick,  guy  their  tops,  raise  the  third  and  brace 
it  to  the  other  two.  A  platform  is  built  on  the  top  of  this  on 
which  the  derrick  is  placed,  another  section  is  then  lifted  into 
place  as  before,  the  derrick  again  moved  up  until  the  top  is 
reached.  Then  the  blocks  are  attached  to  the  top  of  the  tripod, 
which  is  well  guyed,  and  the  sides  of  the  scaffold  raised  as  a 
whole  or  in  sections. 

If  the  station  is  wooded,  one  or  two  large  trees  may  be  left 
standing  and  the  blocks  attached  to  their  tops  for  raising  the 
timbers.  Signals  ninety-four  feet  high  have  had  their  sides  as 
a  whole  put  in  place,  held  there  with  guys  until  the  opposite 
pair  was  raised  and  the  whole  braced  together.  This  can  also 
be  done  in  the  case  of  the  tripod,  by  laying  the  single  piece 
down  with  its  foot  near  its  resting-place,  and  the  pair  lying  in 
the  same  direction  framed  together ;  then  with  ropes  rigged  to 
a  tree  left  standing,  or  to  a  derrick,  the  pair  is  raised  until  it 
stands  at  the  right  inclination,  and  held  in  place  with  ropes 
until  the  single  piece  is  brought  into  position.  To  keep  the 
feet  from  slipping,  an  inclined  trench  can  be  made  towards  the 
hole,  or  they  can  be  tied  to  trees  or  a  stake  firmly  driven  into 
the  ground.  I  have  put  up  tripods  in  a  way  still  different. 
By  framing  one  pair,  and  attaching  between  their  tops  the  top 
of  the  third  by  means  of  a  strong  bolt,  the  whole  stretched  out 
on  the  ground  in  the  shape  of  a  letter  "  Y,"  with  the  feet  of 
the  pair  fastened  near  their  final  resting-place.  The  apex  is 
lifted  and  propped  as  high  as  possible,  then  a  rope  is  passed 


88 


GEODETIC  OPERATIONS. 


through  between  the  legs  of  the  pair  and  attached  to  the  leg  of 
the  single  one  near  its  lower  end.  It  will  be  seen  that  as 
this  leg  is  drawn  towards  the  other  two  the  apex  is  hoisted  up. 

I  have  erected  a  high  signal  in  this  way  by  hitching  a  yoke 
of  oxen  to  the  single  leg  and  hauling  it  towards  the  other  two. 
If  a  tree  should  be  in  a  suitable  place,  a  rope  passing  through 
a  block,  attached  as  high  up  as  possible  in  the  tree,  will  be  of 
great  service  in  hoisting  the  apex. 

A  good  winch  will  be  of  great  use,  and  plenty  of  rope  will 
be  needed,  and  marline  for  lashing.  If  all  the  timbers  are  cut 
and  holes  bored  ready  for  the  bolts,  the  labor  of  erection  will 
be  of  short  duration.  Captain  Boutelle's  tables,  enabling  one  to 
cut  the  timbers  for  a  signal  for  any  height,  are  inserted  here : 
DIMENSIONS  IN  FEET. 


TRIPOD. 

SCAFFOLD. 

Vertical 

Three  feet  below 

Three  feet  below 

S& 
above 
station  • 
point. 

Vert, 
length. 

Slant 
length. 

station-point. 

Vert, 
length. 

Slant 
length. 

station-point. 

Rad.+  o.67 

Side  of  eq. 
triangle. 

One  half 
diagonal. 

Side  of 
square. 

32 

39 

39-31 

5-54 

9.60 

38 

38.52 

14-33 

2O.26 

48 

55 

55-43 

7-54 

13.06 

54 

54-75 

17.00 

24.04 

64 

7i 

71-55 

9-54 

16.52 

70 

70.97 

19.66 

27.80 

80 

87 

87.68 

"•54 

19.99 

86 

87.19 

22.32 

31-57 

96 

103 

103  .  80 

13-54 

23-45 

102 

103.41 

25.00 

35-35 

DIMENSIONS   OF   TRIPOD. 


Slant  dist. 
from  top. 

Vert.  dist.  from 

top  =  L. 

R  =  radius. 
=  j  +  0.667. 

Length  of  hor. 
brace  =  1.732^. 

Length  of 
diagonal 
braces. 

Size  of 
braces. 

Feet. 

Feet. 

Feet. 

Feet. 

Inches. 

O 

o.oo 

0.667 

5 

4.96 

1.287 

2.229 

8 

7-94 

1.659 

2.873 

13 

12.90 

2.279 

3-947 

6.  02 

3  by  2 

20 

19.85 

3.148 

5-452 

8-39 

3  by  2 

29 

28.78 

4.264 

7-385 

11.00 

3  by  2 

40 

39-69 

5.628 

9.748 

13-87 

3  by  2 

53 

52.59 

7.240 

12.540 

17-05 

3  by  3 

63 

67.48 

9-102 

15-765 

20.55 

3  by  3 

85 

84.35 

II.  212 

19.420 

24.40 

3  by  3 

103  .  80 

103  .  oo 

I3.542 

23-455 

25.00 

3  by  3 

FIELD-WORK  OF   THE    TRIANGULATION. 


Ground  
Bottom  of  holes 

Vertical 
length 
from  top. 

Slant 
length 
along 
outside 
edge. 

Feet. 
3-05 
19.27 
35-49 
5I-7I 
67.93 
84.15 
100.38 
103.42 

Slant 
length 
along 
centre 
post. 

Halfdiag. 
from 
station- 
point  to 
outside 
edge. 

Hori- 
zontal 
braces 
side  of 
square. 

Size  of 
hori- 

braces. 

Inches. 
3  by  4 
3  by  4 
3  by  4 
4  by  4 
4  by  4 
4  by  4 

Diag- 
braces. 

Size  of 
diag- 
onal 
braces. 

Inches. 

3  by  3 
3  by  3 
3  by  3 
3  by  4 
3  by  4 
3  by  4 

Feet. 
3 
19 
35 
51 
67 
83 
99 

102 

Feet. 

Feet. 
8.49 
II-I5 
13    82 
16.49 
I9-I5 
21.82 
24.50 
25.00 

Feet. 
12.  OO 
15-77 
19-54 
23.32 
27.08 
30.86 
34-65 
35.36 

Feet. 

21.27 
23-90 
26.81 
29.91 
21.66 
*22.23 

67.50 
83.60 
99.70 
102.72 

' 

*One  foot  from  ground. 

The  floor  of  the  scaffold  should  be  twelve  feet  square,  giving 
room  for  a  tent  large  enough  for  the  observers  to  move  around 
in,  and  sufficient  space  outside  to  pass  around  while  fastening 
the  tent  to  the  railing. 

A  good  shape  for  an  observing  tent  is  hexagonal,  four  and 
a  half  feet  across,  and  six  and  a  half  high,  one  side  opening  for 
its  entire  length  for  exit  and  entrance,  and  the  other  sides  hav- 
ing a  flap  that  opens  from  the  top  to  a  little  below  the  height  of 
the  instrument.  This  will  keep  out  the  sun,  and  also,  by  open- 
ing only  that  part  that  is  needed,  the  tendency  of  the  wind  to 
cool  the  sides  of  the  circle  unequally  can  be  diminished.  A 
corner  post  will  be  needed  at  each  vertex,  and  the  top  can  be 
supported  by  a  rafter  running  from  each  corner  of  the  platform 
meeting  over  the  centre.  To  determine  the  size  of  the  base 
of  the  scaffold,  we  find  the  ratio  of  the  half  diagonal  to  the  ver- 
tical height  and  add  to  this  the  half  diagonal  of  the  top.  One 
foot  in  six  has  been  found  to  give  stability  to  the  signal,  so  that 
for  a  scaffold  80  feet  high  with  3  feet  in  the  ground,  we  have 
for  the  half  diagonal  of  the  base  -8/  -|-  the  half  diagonal  of  the 
top  =  23  feet,  and  the  side  of  the  square  27.6  feet.  The  slope 
can  be  found  by  trigonometry,  tan.  of  slope  =  vertical  height 
divided  by  half  the  difference  of  the  upper  and  lower  diag- 


90  GEODETIC  OPERATIONS. 

onals.  It  is  well  to  brace  the  signal  by  wire  guys  running  from 
each  length  of  timber  in  the  scaffold  legs. 

Probably  the  highest  signal  ever  erected  was  built  by  Assist- 
ant Colonna  of  the  U.  S.  Coast  and  Geodetic  Survey  in  Cali- 
fornia. 

A  large  red-wood  tree  was  cut  off  100  feet  from  the  ground 
and  a  twofold  signal  built, — a  platform  fastened  to  this  high 
stump,  and  a  quadripod  from  the  ground  for  the  support  of  the 
instrument.  The  total  height  was  135  feet.  The  observers 
were  hoisted  up  in  a  chair  attached  to  a  rope  passing  through 
a  fixed  pulley  at  the  top,  and  hauled  by  a  winch  on  the  ground. 

When  the  country  is  approximately  level,  the  curvature  of 
the  earth  will  obstruct  a  long  line  of  sight,  unless  the  instru- 
ment be  elevated  or  a  high  signal  erected.  When  we  know 
the  distance  within  a  mile  or  two  between  the  points  on  which 
it  is  desired  to  establish  stations,  the  problem  is  to  find  how 
high  the  signals  or  scaffolds  must  be  in  order  to  be  intervisible. 
Also,  when  two  suitable  points  of  known  altitudes  are  chosen, 
with  an  intervening  hill  of  known  elevation,  the  problem  is  to 
find  how  high  one  must  build  to  see  over  it. 

Let  h  =  height  in  feet ; 

d  =  distance    of    visibility 

to  horizon  in  feet ; 
R  =  average  radius  of  cur- 
vature in  feet, 
log  R  =  7.6209807. 

The  distance  d  being  a  tangent,  it  is  a  mean  proportional  be- 
tween the  secant  and  the  external  segment,  that  is,  h  :  d  ::  d 
\h  +  2R,  but  h  is  so  small  compared  with  2R  that  it  can  be 
omitted,  and  we  have  k=o.66?2d\  This  is  to  be  increased  by 
its  o.07th  part  for  terrestrial  refraction,  making  h  =0.7139^', 


FIELD-WORK  OF   THE    TRIANGULATION.  9! 

If  we  wish  to  know  how  far  above  the  horizon  the  line  of 
sight  passes  from  two  points  of  known  elevation,  we  find  the 
distance  to  the  point  of  tangency. 

Let  D  —  the  whole  distance  ; 
d  =  the  shorter  distance  ; 
a  —  the  height  above  the  tangent  ; 
m  =  the  coefficient  of  d*  in  the  above  expression. 


k  —  a  =  md\        H—a-  m(D  -  d}*  =  mD1  — 

by  subtraction 

H—h  =  mD*  -  2mDd,         or          2mDd  =  ml?  —  (H  -  K)\ 

therefore,  d 


This  gives  the  distance  from  the  lower  point  to  the  point  of 
tangency  ;  then  the  height  at  which  this  tangent  strikes  either 
station  can  be  found  by  the  above  formula,  h  =  0.7  139^/2,  or 
a  —  h  —0.714^'. 

If  there  is  an  intervening  hill,  we  first  compute  the  point  of 
tangency  of  the  line  from  the  higher  station  ;  then,  how  high 
up  the  intervening  hill  this  tangent  strikes.  To  this  add  the 
amount  by  which  the  lower  hill  exceeds  this  tangent  plane  :  if 
this  be  more  than  the  height  of  the  intervening  hill,  it  can  be 
seen  over  ;  if  less,  the  difference  will  show  how  much  must  be 
added  to  the  height  of  the  terminal  stations. 

If  the  intermediate  hill  be  so  heavily  timbered  as  to  render 
it  impracticable  to  have  it  cleared,  the  height  of  the  trees  must 
be  added  to  the  elevation  of  the  hill  ;  and  at  all  times  it  is  best 
that  the  line  of  sight  should  pass  several  feet  above  all  inter- 
mediate points.  The  following  table  gives  the  difference  be- 
tween the  true  and  apparent  level  in  feet  at  varying  distances  : 


92 


GEODETIC  OPERATIONS. 


Dis- 
tance, 
miles. 

Difference  in  feet  for— 

Dis- 
tance, 
miles. 

Difference  in  feet  for  — 

Curvature. 

Refraction. 

Curvature 
and 
Refraction. 

Curvature. 

Refraction. 

Curvature 
and 
Refraction. 

I 

0.7 

O.I 

0.6 

34 

771-3 

108.0 

663.3 

2 

2-7 

0.4 

2-3 

35 

817.4 

114.4 

703.0 

3 

6.0 

0.8 

5-2 

36 

864.8 

121.  1 

743-7 

4 

10.7 

i-5 

9-2 

37 

9I3-5 

127.9 

785-6 

5 

16.7 

2-3 

14.4 

38 

963.5 

134-9 

828.6 

6 

24.0 

3-4 

20.  6 

39 

1014.9 

I42.I 

872.8 

7 

32-7 

4.6 

28.1 

40 

1067.6 

149-5 

918.1 

8 

42.7 

6.0 

36.7 

4i 

II2I.7 

157-0 

964.7 

9 

54.0 

7-6 

46.4 

42 

II77.0 

164.8 

IOI2.2 

10 

66.7 

9-3 

57-4 

43 

1233-7 

172.7 

1061.0 

ii 

80.7 

"•3 

69.4 

44 

1291.8 

lSO.8 

IIII.O 

12 

96.1 

13-4 

82.7 

45 

I35I.2 

189.2 

1162.0 

13 

112.  8 

15-8 

97-0 

46 

1411.9 

197.7 

1214.2 

14 

130.8 

18.3 

112.5 

47' 

1474.0 

206.3 

1267.7 

15 

150.1 

21.0 

129.1 

48 

1537-3 

215.2 

1322  I 

16 

170.8 

23-9 

146.9 

49 

1602.  o 

224.3 

1377-7 

17 

192.8 

27.0 

165.8 

50 

1668.1 

233-5 

1434-6 

18 

216.2 

30-3 

185.9 

5i 

1735.5 

243.0 

1492.5 

IQ 

240.9 

33-7 

207.2 

52 

1804.2 

252.6 

1551.6 

20 

266.9 

37-4 

229-5 

53 

1874-3 

262.4 

1611.9 

21 

294-3 

41.2 

253-1 

54 

1945.7 

272.4 

1673-3 

22 

322.9 

45-2 

277-7 

55 

2018.4 

282.6 

1735-8 

23 

353-0 

49.4 

303-6 

56 

2092.5 

292.9 

1799.6 

24 

384-3 

53-8 

330.5 

57 

2167.9 

303-5 

1864.4 

25 

417.0 

58.4 

358.6 

58 

2244.6 

314.2 

1930.4 

26 

451-  1 

63.1 

388.0 

59 

2322.7 

325.2 

1997-5 

27 

486.4 

68.1 

418.3 

60 

2402.1 

336.3 

2065.8 

28 

523-1 

73-2 

449-9 

61 

2482.8 

347-6 

2135.2 

29 

561.2 

78.6 

482.6 

62 

2564.9 

359-1 

2205.8 

30 

600.5 

84.1 

516.4 

63 

2648.3 

370.8 

2277.5 

31 

641.2 

89.8 

551-4 

64 

2733-0 

382,6 

2350.4 

32 

683.3 

95-7 

587.6 

65 

2819.1 

394-7 

2424.4 

33 

726.6 

101.7 

624.9 

66 

2906.5 

406.9 

2499.6 

The  following  example  will  illustrate  its  use:  Suppose  we 
have  a  line  of  14  miles  from  A  to  B,  and  at  B  it  is  convenient 
to  build  a  signal  21  feet  high.  By  lobking  in  the  table  in  the 
fourth  column,  we  find  that  the  line  of  sight  will  strike  the 
horizon  at  6  miles,  leaving  8  miles  to  be  overcome  at  A.  Op- 
posite 8  in  the  first  column  we  find  36.7  feet  in  the  fourth, 
therefore  at  A  we  will  have  to  build  37  feet  to  see  B. 

To  illustrate  the  second  problem : 


FIELD-WORK  OF   THE    TRIANGULATION.  93 

Let          h'  =  height  of  higher  station  =  1220  feet; 
h  =  height  of  intervening  hill  =  330  feet ; 
h"  =  height  of  lower  station  =  700  feet; 
d  =  distance  from  h  to  h"  =  24  miles ; 
d'  =  distance  from  h  to  k'  =  40  miles ; 
d-\-  d'  =  distance  from  h'  to  h"  =  64  miles. 

700  feet  strikes  the  horizon  at  34.9  miles,  64  —  34.9  =  29.1 
miles  from  that  point  to  the  other  station.  By  looking  in  the 
table  at  29.1  miles,  the  tangent  strikes  the  other  station  at  486 
feet,  1220  —  486  =  774  feet,  the  distance  the  top  is  above  the 
tangent,  and  29.1  —  24  =  5.1  miles  that  the  point  of  tangency 
is  from  the  intervening  hill,  and  hence  strikes  it  at  15  feet. 
Now,  if  we  conceive  a  line  to  be  drawn  from  the  top  of  the 
higher  to  the  top  of  the  lower,  we  will  have  with  the  tangent 
a  right-angle  triangle  in  which  the  elevations  at  the  higher 
and  intervening  hills  above  the  tangent  are  proportional  to 
their  distances  from  the  lower ;  or,  24  :  64  : :  x  :  774,  x  —  290.2  ; 
that  is,  this  sight-line  strikes  the  intervening  hill  at  290  feet 
above  the  tangent,  and  the  tangent  strikes  it  at  15  feet,  or  the 
sight-line  hits  the  intervening  hill  at  305.2  ;  as  this  is  330  — 
305.2  =  24.8  feet  below  the  top,  the  two  stations  are  not  inter- 
visible. 

The  lower  station  being  the  nearer  the  intervening  hill,  it 
would  be  the  one  to  build  on.  To  determine  the  height  of 
the  necessary  signal,  we  have  the  following  proportion  : 

40  :  64  : :  24.8  :  x,        or        x  =  39.6  feet. 

In  determining  the  altitude  of  stations,  or  intervening  hills, 
an  aneroid  barometer  will  give  a  result  sufficiently  accurate. 
If  the  barometer  is  graduated  to  inches  and  decimals,  the  fol- 
lowing table,  giving  heights  corresponding  to  readings  of  bar- 
ometer and  thermometer,  will  be  useful  in  estimating  the 
height : 


94 


GEODETIC  OPERATIONS. 


Ba- 
rom- 
eter. 

Mean  of  Observed  Temperatures,  Fahrenheit. 

32°. 

42°. 

52°. 

62°. 

72°. 

82°. 

92°. 

30.0 

29.9 

87.5 

89.4 

91.4 

93-3 

95-3 

97-2 

99-2 

29.8 

175-3 

179.2 

183.1 

187.0 

190.9 

194.8 

198.7 

29.7 

263.4 

269.3 

275-1 

280.9 

286.8 

292.7 

298.5 

29.6 

351-8 

359-6 

367-4 

375-2 

383-0 

390-9 

398.7 

29.5 

440.5 

450-3 

460.0 

469.8 

479-6 

489.4 

499.2 

29.4 

529.5 

541.3 

553-0 

564-7 

576.5 

588.2 

600.  i 

29.3 

618.8 

632.6 

646.3 

659-9 

673.7 

687.4 

701.3 

29.2 

708.4 

724.2 

739-9 

755-4 

771-3 

787.0 

802.8 

29.1 

798.3 

816.1 

833-8 

851-3 

869.2 

886.9 

904.7 

29.0 

888.5 

908.2 

927.9 

947-6 

967.4 

987.2 

1007.0 

28.9 

979.0 

1000.7 

1022.4 

1044  .  2 

1065.9 

1087.8 

1109.6 

28.8 

1069.9 

1093-5 

III7-3 

1141.1 

1164.8 

1188.8 

I2I2.6 

28.7 

1161.1 

1186.7 

1212.5 

1238.3 

1264.1 

1290.0 

I3I5-9 

28.6 

1252.5 

1280.3 

1308.1 

1335.9 

1363-8 

I39I-6 

I4I9-5 

28.5 

1344-3 

1374-2 

1404.0 

1433.8 

I463-7 

1493-6 

I523-5 

28.4 

1436-4 

1468.4 

1500.2 

1532.1 

1563-9 

1595-9 

1627.9 

28.3 

1528.5 

1562.9 

1596-8 

163).  7 

1664.5 

1698.6 

1732.7 

28.2 

1621.5 

1657-7 

1693.7 

1729.6 

1765-6 

1801.7 

1837.9 

28.1 

1714.6 

1752.8 

1790.9 

1828.9 

1867.0 

1905-2 

1943-4 

28.0 

1808.  I 

1848.3 

1888.5 

1928.6 

1968.8 

2009  .  o 

2049.3 

27.9 

1901.9 

1944.2 

1986.4 

2028.6 

2071.0 

2113.2 

2155-6 

27.8 

1996.0 

2040.4 

2084.7 

2128.9 

2173-5 

2217.8 

2262.3 

27.7 

2090.5 

2136.9 

2183.4 

2229.6 

2276.3 

2322.7 

2369.3 

•  27.6 

2185.2 

2233.8 

2282.4 

2330  7 

2379-4 

2428.0 

2476.7 

27-5 

2280.3 

2331-1 

2381.7 

2432-2 

2482.9 

2533-6 

2584.5 

27.4 

2375-8 

2428.7 

2481.4 

2534-1 

2586.8 

2639.6 

2692.7 

27-3 

2471.6 

2526.7 

2581.3 

2636.2 

2691.1 

2746.0 

2801.3 

27.2 

2567.8 

2625.0 

2681.9 

2738.9 

2795-9 

2852.9 

2910.3 

27.  i 

2664.3 

2723-6 

2782.6 

2841.8 

2901.0 

2960.2 

30I9-7 

27.0 

2761.2 

2822.6 

2883.9 

2945  •  I 

3006.5 

3067.9 

3I29-5 

When  the  station  is  on  some  hill,  the  name  should  be  the 
popular  designation  of  the  elevation,  or  the  name  of  the  person 
who  owns  the  property  on  which  it  is  situated. 

It  is  a  great  mistake  to  attempt  to  bequeath  to  posterity  the 
name  of  one  of  the  party  locating  the  signal.  When  no  name 
can  be  found  to  attract  attention  to  the  locality  of  the  station, 
a  number  will  answer  the  purpose  of  a  name.  As  soon  as  a 
station  is  ready  for  occupancy  it  will  be  found  advisable  to 
write  a  description  of  the  signal,  its  position,  and  the  way  to 


FIELD-WORK  OF   THE    TRIANGULATION.  95 

reach  it  from  some  well-known  thoroughfare,  to  be  sent  to 
headquarters.  This  would  be  of  especial  service  in  case  the 
work  should  cease  before  the  completion  of  the  observations, 
to  be  resumed  at  some  future  time  by  another  party. 

PORTER. 

"  This  point  is  at  the  head  of  Blue  Lick,  a  tributary  of  the 
Left  Fork  of  Twelve  Pole,  in  Wayne  County,  W.  Va.  It  is 
on  the  farm  of  Larkin  Maynerd.  The  signal  is  built  in  the 
form  of  a  tripod,  and  stands  on  the  highest  point  of  a  large 
field. 

"  A  wagon  can  be  taken  up  the  Twelve  Pole  from  Wayne 
C.  H.,  and  up  Blue  Lick  to  the  signal.  From  this  point  can 
be  seen  Scaggs,  Pigeon,  Williamson,  Vance,  Runyan,  and  Rat- 
tlesnake." 

"Station  No.  24:  Ford  County,  111.;  corner  of  sections,  14, 
15,  22,  23  ;  township,  23  ;  range,  10  east." 

Each  station  should  be  provided  with  an  underground  mark, 
consisting,  when  accessible,  of  a  stone  pier  with  a  hole  drilled 
in  the  top  and  filled  with  lead  bearing  a  cross-mark  on  its  upper 
surface,  the  intersection  forming  the  centre  of  the  station. 
The  top  of  the  stone  should  not  be  within  eighteen  inches  of 
the  surface  of  the  ground,  so  as  to  be  below  the  action  of 
frost,  and  any  disturbance  likely  to  arise  from  a  cultivation  of 
the  soil.  Occasionally  it  has  been  found  convenient  to  build 
above  this  another  pier  to  a  height  of  eighteen  or  twenty  inches 
above  ground,  to  serve  as  a  rest  for  the  instrument  when  the 
station  is  occupied,  and  a  stand  for  the  heliotrope  when  the 
station  is  observed  upon. 

When  large  stone  cannot  be  had,  a  section  of  an  earthen- 
ware pipe  four  inches  in  diameter  may  be  used  by  filling  it 
with  cement  and  broken  stone.  The  upper  surface  can  be 
marked  with  lines  before  the  cement  sets,  or  a  nail  driven  in 
while  it  is  plastic. 


96  GEODETIC  OPERATIONS. 

When  it  is  impracticable  to  dig  a  hole  of  any  depth  for 
a  masonry  superstructure  on  account  of  a  stone  ledge  immedi- 
ately underlying  the  soil,  it  will  be  found  sufficient  to  drill  in 
the  top  of  the  rock  a  hole  and  fill  it  with  lead.  Whatever  form 
of  underground  or  permanent  station-marks  is  used,  it  is  essen- 
tial to  have  surface,  or  reference-marks. 

These  are  usually  large  stones  set  N.,  E.,  S.,  and  W.  of  the 
centre,  and  at  such  distances  that  the  diagonals  joining  those 
at  the  corners  will  intersect  directly  over  the  centre. 

For  immediate  use  it  is  well  to  place  over  the  centre  a  sur- 
face-mark, so  that  should  anything  happen  to  the  signal  before 
it  is  finished  with,  it  can  bq  replaced  without  disturbing  the 
permanent  mark. 

When  the  signal  is  a  high  tripod,  or  when  it  is  necessary  to 
raise  the  instrument  at  the  time  the  station  is  occupied,  the 
relative  position  of  the  centre  of  the  station  and  the  centre  of 
the  instrument  must  be  tested  at  frequent  intervals,  as  an  un- 
equal settling  of  the  signal  would  deflect  it  from  or  towards  the 
centre.  The  quickest  way  to  determine  this  relative  position 
is  to  set  up  a  small  theodolite  at  a  convenient  distance  from 
the  centre,  and  fix  the  intersection  of  the  cross-wires  on  the 
centre  of  the  underground  mark,  or  reliable  surface-mark  ;  then, 
by  raising  the  telescope,  determine  two  points  on  opposite 
sides  of  the  top  of  the  signal.  Then  repeat  this  operation  from 
a  position  approximately  at  right  angles  to  the  first  position. 
Draw  a  string  from  each  pair  of  points  so  fixed,  and  the  inter- 
section of  these  strings  will  indicate  the  centre.  If  possible, 
the  instrument  should  be  placed  directly  over  this  point;  if 
not,  then  the  distance  to  the  point,  and  its  direction  referred  to 
one  of  the  triangle-sides,  should  be  carefully  measured  and  re- 
corded. 

Sometimes  it  happens  that,  in  the  case  of  a  very  high  signal 
situated  on  a  sharp  point,  no  position  can  be  found  from  which 
both  the  top  and  the  station-mark  can  be  seen.  To  meet  just 


FIELD-WORK  OF   THE    TRIANGULATION.  97 

this  difficulty,  Mr.  Mosman  has  devised  an  instrument  with  a 
vertical  axis  resting  on  levelling-screws,  and  so  adjusted  that 
when  freed  from  errors  the  telescope  revolves  around  an 
imaginary  axis  passing  through  the  intersection  of  the  cross- 
wires. 

The  optical  features  are  such  as  to  admit  of  focusing  it  on 
objects  at  distances  varying  from  a  few  inches  to  150  or  200 
feet.  To  use  it,  you  place  it  on  a  support  over  the  centre  of 
the  station,  the  support,  of  course,  having  a  hole  through  it. 
After  levelling  the  instrument,  move  it  until  the  cross-wires 
coincide  with  the  station-mark ;  then,  by  simply  changing  the 
focus,  a  point  can  be  found  in  the  intersection  of  these  wires. 
This  operation  should  be  repeated  with  the  telescope  in  dif- 
ferent positions;  and  as  different  points  are  obtained,  the  centre 
of  the  figure  formed  by  joining  these  points  will  be  the  one 
desired. 

The  reverse  operation  can  also  be  successfully  performed 
with  this  instrument. 

It  sometimes  occurs  that  a  straight  tree  is  used  as  a  signal, 
in  which  event  it  is  necessary  to  occupy  an  eccentric  station. 
This  must  be  marked  with  as  much  care  as  though  it  were  the 
true  station.  The  method  for  reducing  the  observed  angles 
will  be  given  in  full  on  page  143. 

In  the  record-book  must  be  kept  a  description  of  the  mark- 
ings of  the  stations  ;  and  when  an  eccentric  position  is  occupied, 
the  distance  and  direction  already  referred  to  are  to  be  carefully 
entered. 

The  method  of  observing  horizontal  angles  must  depend 
upon  the  accuracy  desired  and  upon  the  kind  of  instrument 
used.  Regarding  the  maximum  error  in  closing  primary  tri- 
angles to  be  three  seconds,  or  six  for  secondary,  a  number  of 
precautions  must  be  taken.  The  principal  ones  may  be  classed 
under  the  following  heads : 
7 


g8  GEODETIC  OPERATIONS, 

1.  Care  in  bisecting  the  object  observed  upon. 

2.  Stability  of  the  theodolite-support. 

3.  Elimination  of  instrumental  errors. 

4.  Preservation  of  the  horizontality  of  the  circle. 

5.  Rapidity  of  pointings. 

6.  Observations  at  different  times  of  the  day. 
Conditions  I  and  2  are  self-apparent,  and  the  best  means  of 

compliance  therewith  will  readily  suggest  themselves  to  the 
observer. 

The  instructions  for  eliminating  instrumental  errors  have 
already  been  given  (see  Chap.  II.). 

When  the  theodolite  is  placed  in  position  and  levelled,  see 
that  the  adjustments  have  not  been  disturbed  before  beginning 
a  set  of  readings.  If,  while  observing,  the  level  shows  a  change 
in  the  horizontality  of  the  circle,  do  not  disturb  it  until  the 
set  is  finished.  But  if  the  deflection  be  considerable,  the  read- 
ings must  be  thrown  away. 

The  advantage  of  pointing  rapidly  is  the  greater  certainty 
of  having  the  same  state  of  affairs  when  sighting  to  all  of  the 
signals  in  the  circuit,  since  it  diminishes  the  interval  during 
which  there  can  occur  unequal  expansion  of  the  circle ;  twist 
in  the  theodolite-support,  changes  in  the  illumination  of  the 
different  signals,  or  flexure  of  the  circle  from  any  cause. 

By  making  observations  at  different  times  of  the  day,  errors 
arising  from  lateral  refraction  may  be  diminished  because  of 
the  changes  in  the  condition  of  the  atmosphere. 

There  are  two  principal  classes  of  theodolites — repeating-  and 
direction-instruments.  The  former  gives  a  number  of  readings 
in  a  short  time,  but  a  new  source  of  errors  is  introduced  by  the 
repeated  clamping  and  unclamping.  However,  if  the  clamps 
do  not  produce  what  is  called  travelling,  the  principle  of  repeti- 
tions renders  it  possible  to  obtain  a  large  number  of  readings 
on  all  parts  of  the  circle,  and  thus  tends  to  free  the  average 
from  the  effect  of  errors  of  graduation,  for  if  the  divisions  on 


FIELD-WORK  OF   THE    TRIANGULATION.  99 

one  side  of  the  circle  are  too  far  apart,  there  will  be  other 
parts  on  which  the  divisions  are  too  close.  In  measuring  an 
angle  with  a  repeater,  it  is  best  to  set  the  circle  at  zero  ; 
point  on  the  first  station  on  the  left,  bisect  the  signal,  see  that 
the  circle  is  clamped,  and  then  turn  to  the  next  station.  Read 
and  record  both  verniers,  turn  the  entire  instrument  back  to 
the  initial  point  and  bisect ;  then  unclamp  the  telescope  and 
point  to  the  second  station,  clamp,  and  turn  back  to  the  first. 
Repeat  this  operation  until  the  whole  circle  has  been  passed 
over;  divide  the  last  readings  by  the  number  of  pointings,  and 
the  quotient  will  be  the  value  to  adopt  as  the  average  for  the 
two  verniers.  The  advantage  of  recording  the  first  reading  is 
that  it  serves  as  a  check  on  the  number  of  degrees  and  minutes 
in  the  final  result. 

If  there  are  several  angles  at  a  station,  it  is  advisable  to  read 
them  individually  and  in  all  combinations.  Calling  the  angles  in 
their  order,  I,  2,  3,  4,  and  5,  we  read  and  repeat  I,  then  2, 
3,  4,  and  5  ;  afterwards  I,  2,  as  one  angle;  then  I,  2,  3,  as  one ; 
I,  2,  3,  4,  as  one  ;  and  I,  2,  3,  4,  5  ;  also  2,  3  ;  2,  3,  4  ;  and  2, 
3,  4,  5  ;  then  3,  4;  3,  4,  5  ;  and,  finally,  4,  5,  this  closing  the 
horizon.  The  advantage  of  this  can  be  seen  when  we  take  up 
the  adjustment  of  the  angles  around  a  station.  When  an  in- 
strument can  be  reversed  in  its  AT's,  it  will  be  found  desirable 
to  make  a  similar  set  with  the  telescope  reversed,  and  record 
these  as  R. 

With  a  direction-instrument,  it  is  not  necessary  to  make 
these  combinations.  The  plan  is  to  make  5,  7,  II,  13,  17,  19, 
or  23  series,  by  dividing  the  circle  into  such  a  number  of  parts; 
as  each  one  is  prime  to  two  or  three  reading-microscopes,  no 
microscope  can  fall  on  the  same  part  of  the  limb  twice  in 
measuring  the  same  angle. 

Suppose  we  decide  to  make  eleven  series,  we  first  find  the 
initial  pointing  for  each  set  of  the  series.  One  eleventh  of 
360°  =  32°  43'  38".2,  two  elevenths  =  64°  27'  i6".4,  etc. 


IOO  GEODETIC  OPERATIONS. 

We  set  the  circle  approximately  on  zero,  and  turn  the  entire 
instrument  upon  some  arbitrary  point  that  can  be  readily  bi- 
sected, cla.mp  on  this,  and  make  bisection  perfect  by  moving 
tangent-screw  of  telescope  if  necessary.  Read  and  record  all 
the  microscopes,  taking  both  forward  and  backward  microme- 
ter-readings as  already  explained.  Turn  the  telescope  until  the 
next  signal  is  bisected  ;  read  and  record  as  before ;  continue 
around  to  the  last.  After  recording  this  last  reading,  see  if  the 
signal  is  bisected ;  if  so,  record  the  same  values  as  just  read  for 
the  first  on  the  return  set.  When  the  initial  point  is  reached, 
reverse  the  telescope  and  make  a  set  as  before,  recording  this  set 
as  R.  Then  set  on  the  second  position  and  make  a  complete 
set,  continuing  in  this  way  until  all  of  the  positions  are  used. 

It  is  desirable  to  sight  on  all  of  the  signals  every  time,  and 
also  on  the  azimuth-mark  if  one  has  been  erected  ;  but  if  one 
should  become  indistinct,  while  all  the  others  show  well,  this 
one  can  be  omitted  and  supplied  afterwards.  It  will  be  seen 
at  once  that  by  this  method  we  get  an  angle  as  the  difference 
of  two  directions ;  hence  the  probable  error  of  an  angle  will  be 
\^2  times  the  probable  error  of  a  direction.  The  record-book 
should  be  explicit,  giving  the  time  of  each  pointing,  position 
of  circle  at  the  initial  point,  the  position  of  the  telescope,  D  or 
R,  appearance  of  signal  (the  latter  is  of  importance  in  weighting 
angles);  also, if  a  tin  cone  is  sighted,  the  time  and  direction  of 
the  sun  referred  to  the  cone  must  be  recorded  as  data  for  cor- 
recting for  phase. 

If  the  triangulation  is  for  general  topographic  purposes,  it 
will  be  found  advisable  to  read  angles  to  prominent  objects 
that  may  be  in  view,  since  if  one  is  seen  from  two  well-deter- 
mined stations  its  position  can  be  approximately  located. 
Preliminary  computations  should  be  carried  along  in  the  field, 
so  as  to  apply  reduction  from  eccentric  stations  to  centre  and 
deduce  the  probable  errors  of  the  angles. 


FIELD-WORK  OF   THE    TRIANGULATION. 


101 


If  this  falls  beyond  the  predetermined  limit,  ortif  the  tri- 
angles do  not  close  after  allowing  for  spherical  excess  within 
the  limit  prescribed,  the  angles  should  be  remeasured. 

The  example  here  given  is  taken  from  record-book  just  as  it 
came  from  the  field — d  is  the  forward,  and  d'  the  backward 
micrometer-reading : 


HORIZONTAL   DIRECTIONS. 


Station:  Holmes,  W.  Va. 
Observer:  A.  T.  M. 


Instrument:  114. 


Date:  Sept.  7,  1881. 
Position:  n. 


Series 
and 
No. 

Objects 
Observed. 

Time. 
h.    tn. 

Tel. 

Mic. 

' 

' 

d. 

d'. 

Mean 
d. 

Remarks. 

io 

Table  Rock.  .  .  . 

5  '•  io 

D 

A 

32 

44 

48 

48 

Weather  clear,    at- 

B 
C 

25 
25 

*4 

32.50 

mosphere   moder- 
ately clear. 

Somerville  

5  =  8 

D 

A 
B 

319 

44 

74 

5" 

73 
49 

Wind    S.W.,    light. 
Ther.  97«.5. 

C 

53 

53 

58.66 

Somerville  .... 

5  -  '3 

R 

A 

139 

44 

73 

7« 

Reversed. 

B 

40 

;  } 

C 

48 

47 

52.83 

Table  Rock.... 

5  :  20 

R 

A 

g 

212 

44 

5° 

5° 

C 

30 

ag 

33-o 

X1 

Table  Rock.... 

5  :  35 

R 

A 

•13 

44 

49 

48 

B 

IO 

C 

3° 

28 

32.16 

Piney  

5:33 

R 

A 

301 

18 

64 

«3 

Heliotrope. 

B 

34 

C 

45 

43 

47-33 

Reversed. 

In  addition  to  the  determination  of  the  geographic  positions 
of  various  points  by  triangulation,  it  is  also  possible  to  obtain 
with  some  precision  their  elevation,  for,  since  we  compute  the 
distances  between  the  stations,  the  only  remaining  term  is  the 
angle  of  depression  or  elevation  from  the  station  occupied  to 
each  that  can  be  seen.  This  necessitates  a  determination,  by 
levelling,  of  the  height  of  the  initial  point  only. 

The  field-work  can  be  easily  described  as  consisting  of  a 


I02  GEODETIC  OPERATIONS. 

number  of  readings  of  the  angles  from  the  zenith  to  each  sta- 
tion. In  the  computation  given  on  page  90,  it  will  be  seen 
that  vertical  refraction  affects  this  angle ;  but  if  the  zenith-dis- 
tances be  measured  from  each  station  to  the  others  at  the 
same  time,  supposing  the  refraction  to  be  equable  throughout 
the  intervening  space,  the  uncertainty  caused  by  the  unknown 
deflection  of  the  sight-line  will  be  eliminated. 

But  as  it  is  not  always  feasible  to  have  all  the  points  oc- 
cupied at  the  same  time,  the  zenith-distances  can  be  meas- 
ured on  different  days,  and  when  possible,  under  such  varying 
atmospheric  conditions  as  to  secure  the  same  average  relative 
refraction.  The  best  time  is  between  the  hours  10  A.M.  and 
3  P.M.  The  height  of  the  theodolite  above  ground  must  be 
known,  as  well  as  that  of  the  signal  sighted. 

In  1860,  Assistant  Davidson  organized  a  series  of  experi- 
ments to  obtain  a  comparison  of  the  various  methods  of  deter- 
mining altitudes.  He  used  a  Stackpole  level,  a  rod  carefully 
compared  with  a  standard  and  levelled  in  both  directions.  The 
measures  of  zenith-distances  were  reciprocal.  They  were 
made  seven  times  daily  for  five  days.  The  barometric  series 
consisted  of  hourly  readings  during  five  days  of  a  mercurial 
barometer,  attached,  detached,  and  wet-bulb  thermometers. 
The  differences  in  the  altitudes  are  : 

As  determined  by  levelling,  598-74  metres. 

"  "  zenith-distances,  598.64       " 

"  "          atmospheric  pressure,  595.26       " 

REFERENCES. 

U.  S.  Coast  and  Geodetic  Survey  Reports,   1876,  pp.  238- 
401 ;  '80,  pp.  96-109;  '82,  pp.  151-208. 
Puissant,  G£ode"sie,  vol.  i.  pp.  350-376. 
Bessel,  Gradmessung  in  Ostpreussen,  pp.  59-128. 


FIELD-WORK  OF   THE    TRIANGULATION.  IO3 

Ordnance  Survey,  Account  of  the  Principal  Triangulation, 
pp.  1-61. 

Struve,  Arc  du  M£ridien,  vol.  i.  pp.  1-35. 

Publications  of  the  Prussian  Geodetic  Institute,  especially 
"  Das  Hessische  Dreiecksnetz"  and  "  Das  Rheinische  Dreiecks- 
netz,"  II.  Heft. 


IO4  GEODETIC  OPERATIONS. 


CHAPTER  V. 

THEORY   OF   LEAST   SQUARES. 

WHEN  in  the  various  measurements  of  a  magnitude  a  num- 
ber of  results  are  obtained,  it  is  a  matter  of  great  importance 
to  know  which  to  regard  as  correct.  That  all  cannot  be  cor- 
rect is  apparent,  and  that  some  one  is  true  may  safely  be 
assumed.  Errors  which  render  a  magnitude  too  great  are 
called  negative  errors,  and  those  which  make  it  too  small 
are  positive  errors.  Should,  for  instance,  the  true  length  of  a 
line  be  73.45  chains,  and  its  length  found  by  measurement  to 
be  73.44  chains,  the  error  would  be  -j-  O.OI  chain  ;  while  if 
the  measurement  show  73.46  chains  the  error  would  be  —  o.oi 
chain. 

It  may  be  accepted  as  a  general  rule  that  positive  and  nega- 
tive errors  are  equally  probable ;  also,  that  small  errors  are 
more  likely  to  occur  than  great  ones,  since  the  tendency  to 
commit  a  great  error  would  be  readily  detected  before  record- 
ing it,  while  those  smaller  could  not  be  easily  distinguished 
from  the  value  afterwards  found  to  be  correct. 

Let  the  angle  x  be  measured  n  times  with  equal  care,  so 
that  in  each  result  there  is  the  same  liability  for  an  error  to 
occur;  let  the  individual  values  obtained  be  vvvv...vn. 
Since  x  is  the  true  value,  the  errors  will  be :  x  —  #,,  x  —  z/2, 
.  .  .  x  —  vn;  these  we  will  denote  by  dxr  dxv  .  .  .  dxn,  and, 
from  what  has  just  been  said,  some  are  positive  and  some  are 
negative.  As  there  exists  the  same  probability  for  the  posi- 
tive as  for  the  negative  errors,  and  since  the  individual  errors 


THEORY  OF  LEAST  SQUARES.  10$ 

are  nearly  equal  to  each  other,  their  sum  will  nearly  amount 
to  zero,  and  we  may  put, 


dx,  +  dx,  +  dx*  +  dXi+...  +  dxn  =  o, 
or  (x  -  v,}  +  (x  -  vt)  .  .  .  +  (x  -  vn}  =  o, 

whence         nx  =  z/,  -f-  v^  -f-  vt  -f-  .  .  .  -f~  vn.     Hence  : 

(i)  x  =  (vl  -f-  z>3  +  ^3  +  •  •  •  v«)  -T-  «,  which  is  simply  the 
arithmetical  mean  of  the  n  terms.  This,  however,  gives  no  infor- 
mation as  to  the  value  of  the  errors.  If  for  each  positive  error 
we  had  committed  an  equal  negative  error,the  arithmetical  mean 
would  give  the  correct  value,  but  this  fortuitous  elimination 
can  only  be  expected  in  an  infinite  number  of  observations  ; 
even  then  it  will  not  enable  us  to  form  any  definite  opinion  as 
to  the  degree  of  accuracy  attained  in  the  individual  observa- 
tions. In  order  to  accomplish  this,  we  must  find  some  means 
for  preventing  the  positive  errors  from  destroying  the  negative 
ones.  Gauss  found  the  way  by  taking  into  account  not  the 
errors  themselves,  but  their  squares,  which  are  positive,  and 
hence  cannot  eliminate  one  another. 

For  brevity  we  will  write  [zy]  for  the  series  of  terms  involv- 
ing v,  as  vt,  vy,  va,  .  .  .  vn,  and  S[vM~]  for  the  sum  of  such  a  series. 
Hence  we  may  put  for  the  sum  of  the  squares  of  the  errors, 
S\d,tx~^,  or  S[;tr  —  vjf.  The  value  of  x  will  approach  the 
nearest  to  its  correct  value  when  the  arithmetical  sum  of  the 
errors  is  the  smallest,  or  when  the  sum  of  the  squares  of  the 
errors  is  a  minimum  ;  that  is,  when  5  [^«^2]  is  a  minimum. 

Let  y 

then  by  differentiation,-^-  =  2S[dnx~\. 


IO6  GEODETIC  OPERATIONS. 

As  this  is  to  be  a  minimum,  we  place  the  first  differential  co- 
efficient =  o, 

or  S\dnx\  =  o,         or        S[(x  —  *<)]  =  o. 

If  x  —  v,  +  x  —  v^  +  x  —  v3  .  .  .  -f  x  —  vn  —  o, 


or 


a  result  identical  with  the  one  previously  obtained. 

The  converse  can  also  be  demonstrated;  that  is,  the  arithmet- 
ical mean  gives  to  the  square  of  the  residuals  the  minimum, 


\dnx\  = 

=  (*  -  v,}  +  (x  -  Vf)  +  (x  -  v^  .  .  .  +  (x  -  VH)  =  o  ; 


squaring  this, 

r  =  (*  -  v>r  +  (*  -  »,)• 


but  x  =  *2S*         and 

substituting  these  values, 


'vj?  =  ff.T  —  JJLL        .     (0 
-  "J        L  "J  w  k  ; 


Suppose  we  now  take  some  other  value,  JF,,  so  that  d^x^  d^x^ 
.  .  .  dnxl  represent  the  residuals,  then 


THEORY  OF  LEAST  SQUARES. 


substituting  in  this  equation  the  value  for  [z/«]z  derived  from 


therefore,  [<W  =  \drtxj  -f  n(x  —  x$  .....     (2) 


Since  (x  —  x$  is  always  positive,  \dnx^  is  greater  than 
that  is,  the  square  of  the  residuals  when  any  value  other  than 
the  arithmetical  mean  is  used,  is  greater  than  when  the  arithmet- 
ical mean  is  taken.  From  this  it  is  seen  that  the  arithmetical 
mean  is  the  most  probable  value  ;  but  the  correct  value  might 
be  a  little  more  or  a  little  less  than  this  mean. 

When  the  individual  results  are  nearly  the  same,  we  might 
be  satisfied  with  any  one  ;  but  when  there  is  a  great  range,  we 
accept  the  average  even  with  some  trepidation.  Now,  if  we 
had  some  term  that  depended  upon  the  residuals,  the  magni- 
tude of  this  term  might  be  taken  as  the  measure  of  precision  : 
this  is  what  is  known  as  the  probable  error. 

The  development  of  an  expression  for  the  probable  error 
has  been  undertaken  by  many  persons,  and  prosecuted  in  vari- 
ous ways,  but  all  attaining  the  same  end.  The  discussion  that 
follows  is  taken  principally  from  Chauvenet. 

Let  us  recapitulate  what  are  known  as  the  theorems  of  the 
theory  of  probabilities. 


108  GEODETIC  OPERATIONS. 

1st.  Equal  positive  and  negative  errors  are  equally  probable, 

"\,  in  a  large  series  of  observations,  are  equally  frequent. 

2d.  There  is  a  limit  of  error  which  the  greatest  accidental 
errors  do  not  exceed  ;  if  /  denote  the  absolute  magnitude  of 
this  limit,  all  the  positive  errors  will  be  comprised  between  o 
and  -j-  ^  ar>d  all  the  negative  errors  between  oand  —  /,  so  that 
the  errors  are  contained  within  2/. 

3d.  Small  errors  are  more  frequent  than  large  ones. 

So  that  the  frequency  of  the  error  may  be  considered  a  func- 
tion of  the  error  itself.  If  A  be  an  error  of  a  certain  magnitude, 
and  its  frequency  cpA,  this  function  will  be  a  maximum  when 
A  =  o,  and  be  o  when  A  =  ±  /.  If  we  denote  the  probability 
of  an  error  A  by  y,  we  have  y  =  (pA,  an  equatioij  of  a  curve 
in  which  A  is  the  abscissa  and  y  the  ordinate;  as  A  has  equal 
values  with  contrary  signs,  the  curve  is  symmetrical  with  re- 
spect to  the  axis^,  and  for_^  =  o,  A  =  +00. 

We  shall  therefore  consider  A  as  a  continuous  variable,  and 
cpA  as  a  continuous  function  of  it. 

If  there  are  n  errors  equal  to  A,  n'  =  A'  .  .  .  ,  and  the  entire 
number  equal  to  m,  the  respective  probabilities  are 


n  n 

a>A  =  — ,         q>A  •=.  — ,  etc., 
m  m 


therefore,  <pA  -\-  <pA'  -J-  <pA"  -j-  .  .  .  =  i. 


However,  the  continuity  of  the  curve  requires  that  the  suc- 
cessive values  of  A  shall  differ  from  one  another  by  an  infini- 
tesimal, so  that  the  number  of  values  for  q>A  is  infinite. 


THEORY  OF  LEAST  SQUARES.  IOQ 

Let  us  take  the  smallest  unit  of  magnitude  in  the  observa- 
tions as  I,  then  the  probability  of  the  error  A  maybe  regarded 
as  the  same  as  the  probability  that  the  error  falls  between  A 
and  A  -\-  i,  and  the  probability  of  an  error  between  A  and 
A  -\-  i  will  be  the  sum  of  the  probabilities  of  the  errors  A, 
d  _|_  It  A  4-  2  .  .  .  A  4-  (i  —  i).  By  making  i  small  the  proba- 
bility of  each  of  the  errors  from  A  to  A  -\-  i  will  be  nearly  the 
same  as  A,  and  their  sum  will  approximate  i<pA.  When  the 
interval  between  the  successive  errors  approximates  an  infini- 
tesimal, the  expression  becomes  more  nearly  exact,  and  for  i 
we  may  put  dA,  and  write  (pA  .  dA  as  the  accurate  expression 
for  the  probability  that  an  error  falls  between  A,  and  A  -f-  dA. 
Hence  the  probability  that  an  error  falls  between  the  limits 
-f-  oo  and  —  oo  is  the  sum  of  the  elements  of  the  form  cpA .  dA, 

or  the  integral  /       cpA  .  dA  =  i. 

Suppose  the  quantity  M  be  a  function  of  x,  y,  z,  etc.,  A,  A', 
A" ,  be  the  errors,  and  <pA,  cpA',  <pA" ,  their  respective  probabili- 
ties. 

Since  the  probability  of  J/will  be  the  product  of  the  proba- 
bilities of  the  quantities  of  which  Mis  a  function,  we  may  put 
P  =  cpA  .  (pA' .  cpA"  .  .  . 

From  preceding  principles  we  know  that  the  most  probable 
system  of  values  of  the  unknown  quantities  x,  y,  z  .  .  .  will  be 
that  which  makes  Pa.  maximum  ;  therefore  we  obtain  the  dif- 
ferential coefficient  of  P  with  respect  to  each  variable  and 
place  it  equal  to  o.  Log  P  varies  with  P,  and  as  P  is  a  func- 
tion of  x,  y,  2,  .  .  .  ,  the  differential  coefficients  of  P  with 
respect  to  x,  y,  z,  .  .  .  ,  must  separately  =  o ;  or 


I     dP  I     dP  I     dP 

'         =  °'  --=°>  '=°' 


But,         log  P  =  log  (pA  -f  log  <pA'  -\-  log  (pA"  ...  (2) 


HO  GEODETIC  OPERATIONS. 

dP      dcpA    .    dtpA'    .   dcpA" 
Therefore,     _  =  -^  +  -^-  +        /T.  .  .  (3) 


Divide  (3)  by  dx,  dy,  and  dz,  ... 


•(A) 


dP 

~  q>Adx 
dcpA 

dcpA'     }     dcpA" 

P.dx 

dP 
PTTy 

dP 

TTdz 

cpA'dx       (pA"dx 
d(pA'          dcpA" 

~  cpAdy 
d(pA 

d(pA' 

dcpA" 

etc., 

<pA'dz 

\    <pA"dz  °» 
etc.; 

since  the  first  members  are  equal  to  zero,  from  (i). 


In  (A)  let  us  place  for  — -: ,  cp'A.  dA;  for  -~T,  <P'A' .  d  A'. 
tpA  q)A 

Then  they  will  become 


etc.,  etc. 

If  *  be  the  correct  value  of  M,  M,  M",  etc., 
A  =  M-x,         A'=M'-x,        A"  = 


THEORY  OF  LEAST  SQUARES.  Ill 

dA         dA'         dA" 

from  which  -7-  =  —r-  =  — r—  .  .  .  =  —  i. 

dx        dx         dx 


The  first  equation  at  (B)  becomes 

i 
-x)  +  9'(M'  -x)  +  cp'(M"  -*)...  =  a      (4) 


Now,  if  in  this  equation  we  suppose  M'  =  M"  .  .  .  =  M 
—  mN,  where  m  represents  the  number  of  observations,  and 
since  the  arithmetical  mean  is  the  most  probable  value  of  x, 


4-M"  ... 


since  there  will  be  m  —  I  terms  after  the  first,  each  equivalent 
to  M  —  mN, 

or  x  =  ~(M  +  mM  -  nfN  -  M+  mN) 


=  -(mM  -  m 


=  M  —  mN  +  N 
=  M  —  (m  —  i)N; 

or  M  —  x  =  (m  —  i)N, 

and  M'  -  x  -  M"  -  x  =  M  -  mN  -  X-, 


H2  GEODETIC  OPERATIONS. 

but  x  =  M  —  mN  +  N\ 

therefore 


=  M  -  mN  -  M  +  mN  -  N  =  -  N. 
Substituting  these  values  for  x  in  (4),  we  get 

<p'  [(m  -  i)N]  +  (m  -  !)?'(-  N)  =  o, 

since  there  are  m  —  I  terms  after  the  first  each  equal  to  —  N. 
Transposing, 


dividing  by  N, 

cp'(-  N) 


(m-  \}N  -  N   ' 

This  is  a  true  expression  for  all  values  of  m,  or  (m  —  i),  or 
N(m  —  i).  As  the  second  term  is  not  affected  by  changes 
in  m,  the  expression  is  a  constant.  By  putting  A  =  (m  —  i)N, 

cp'A 
we  will  have  —-  =  a  constant  which  is  called  K. 


If  —-  =  K,         cp'A  = 


THEORY  OF  LEAST  SQUARES.  113 


dcpA 
But  we  supposed  cp' A  =  —  y      -  (page  1 10); 


theref°re 


- 
cpA 


Integrating  this,      log  cpA  =  \KA*  -f-  log  c, 
or  <pA  =  C(*Kl^, 

in  which  e  is  the  Naperian  base.  • 

Since  (pA  must  decrease  as  \K  increases,  it  must  be  negative. 
Placing  —  h*  for  \K,  we  have 


likewise  <p  A'  =  ce  ~  h^'\ 

cpA"  =  ce-w\ 
etc. 

We  found  that    P  —  <pA  .  <pA'  .  <pA"  .  .  .  ; 

therefore  P  =  c(e~  h*<*  -\-  e~  *'A"  +  <r  »*'*  .  .  .), 

/>=<,--';    ......    V*     .     (5) 

which  represents  the  probabilities  of  all  errors  from  A  to  A* 
inclusive. 
8 


H4  GEODETIC  OPERATIONS. 

To  determine  the  constant  c,  we  will  take  the  integral  on 
page  109, 


f      cpA 
and  substitute  for  (pA,  ce~  *'A*  , 


We  will  write      ?  =  f?A\        and        A  =  v  ; 

/»+«  / 

then  we  obtain  /       ce~fld-r\ 

J-  »  k 

factoring  |  ,  j£  m  e-*dt=i. 

Let  m=f     die-**; 

»/—  00 

then,  since  this  integral  is  independent  of  the  variable,  we  may 
also  put 

m  =  f     due-  «'; 

J-o> 

by  multiplication, 

***.&*-&+*>  .....     (6) 


THEORY  OF  LEAST  SQUARES.  11$ 

If  we  integrate  between  the  limits  oo  and  o,  then  between 
the  limits  o  and  —  oo  their  sums  will  be  the  value  of  the  defi- 
nite integral  between  the  limits  -j-  oo  and  —  oo. 

We  shall  now  place  u  —  tv,  and  du  =  tdv ;  then  (6)  becomes 


wt  =  dv  .dt.te- 


v  being  regarded  as  the  variable,  and  t  the  constant, 


=SJdv  •  iF^  =  Ktan  "  T  °°  "  tan  "  I 


w  — 


Without  changing  notation  : 


or  the  total  integral,     _^  dte~*=  Vx 


But 


Il6  GEODETIC  OPERATIONS. 


therefore         r  VTC  —  I,         c1/n  =  hy         c=  —=-. 
n  \  n 


Placing  this  value  for  c  in  tpA  =  ce  ~  h"^ 


we  have  <pA  =  —;=e  ~  ASA* . 


This  method  of  obtaining  the  value  for  the  definite  integral 
is  taken  from  Laplace,  Mfchanique  Celeste. 

Identical  results  are  obtained  by  different  methods  by  Pois- 
son  and  Airy. 

The  probability  that  the  error  falls  between  A  and  A  -\-  dA 

h 
is  —  =*-**\W,  and  that  it   falls  between  the  limits  o  and  a  is 

Vn 

h     f*  =  * 

-  /         e-h*^dA,  as  already  explained. 

t/A  =  o 


Let     t  —  hA,      then     A  —  T;    and  a?    —  A  y,      then     t  =  ah. 
h  li 

Substitute  these  values  in  the  last  integral  ;  it  becomes 


after  multiplying  by  two,  since  the  sum  of  negative  errors  is 
considered  equal  to  the  positive  errors. 

This  integral  has  been  computed  for  values  of  t.  A  table  of 
these  is  given  in  Merriman's  "  Least  Squares." 

From  this  table  it  is  found  that  the  error  which  occupies  the 
middle  place  in  the  series  of  errors,  arranged  in  the  order  of 


THEORY  OF  LEAST  SQUARES.  IT/ 

their  magnitude,  has  the  same  number  of  errors  above  as  be- 
low ;  therefore,  the  error  satisfying  this  condition  is  that  for 
which  the  value  of  the  integral  is  £.  If  we  designate  the  cor- 
responding value  of  /  by  p,  we  find  from  the  same  table  that 


p  =  0.4769. 


If  r  be  the  error  in  a  series  of  observations  whose  precision  is 

P  ,  P 
T»  h  —  - 
h  r 


P  P 

h,  we  can  put  p  =  hr,  r  =  j,  h  =  —. 


MEAN  OF  THE  ERRORS. 

If  we  have  a  series  of  m  errors  A,  A'  .  .  .\  a  positive,  and  a 
negative  each  equal  to  A,  or  2a  in  all,  the  probabilities  are  that  in 

all  there  will  be  — ,  —  .  .  .     The  mean  of  these  errors,  suppos- 
m    m 

ing  each  repeated  a  number  of  times  proportional  to  the  proba- 

.    2aA  +  2a'A'  +  2a"A" 
bihty  of  its  occurrence,  is .  .  . 


=  2J—  +  2A'— 
m  m 


The  probability  of  an  error  A  has  been  shown  equal  to  <pA  .  dA. 
So  that  the  above  expression  for  an  infinite  number  of  terms 
approximates  a  series  of  terms  of  the  form  2 A  .  cpA  .  dA.  But 
on  page  1 16 


Il8  GEODETIC  OPERATIONS. 

multiplying  by  2  A  .  dA,  we  have 


.  dA  =  -,Ae- 

'    7t 


If  17  be  the  mean  error, 


h——\        therefore          rj  =  -  =  -  1=-  —  I.i82or; 

r  P     - 


or  r  =  0.84537,         since        p  —  0.4769 

As  was  stated  elsewhere,  it  is  not  feasible  to  obtain  the 
mean  of  the  error,  since  the  negative  and  positive  errors  being 
theoretically  equal,  their  sum  will  become  zero.  So  we  take 
the  sum  of  the  squares  of  these  errors,  and  the  square  which  is 
the  mean  of  these  squares  is  the  square  of  the  mean  error. 

That  is,  if  s  be  the  mean  error, 


=  -1; 

2/*" 

r 


hV2          P_ 

r 
r  =  0.6745  s. 


=  i.4826r  ; 


This  value  of  r  is  the  probable  error  of  anyone  of  the  observed 
values  of  the  unknown  quantity,  x. 


THEORY  OF  LEAST  SQUARES.  1 19 

Let  us  now  look  for  an  expression  for  the  value  of  the 
arithmetical  mean,  ra. 


Equation  (5),  P  —  <pA  -j-  tpA'  -J-  cpA"  .  .  . 


The  most  probable  value  of  the  observed  quantity  is  that 
which  makes  Pa.  maximum,  or  that  makes  A*  -j-  A'*  -|-  A"*  .  .  . 
a  minimum. 

But  it  has  been  shown  that  the  arithmetical  mean  renders 
the  sum  of  the  squares  a  minimum  ;  therefore  P  represents  the 
probability  of  the  arithmetical  mean  when  A,  A',  A"  .  .  .  repre- 
sent the  residuals  referred  to  this  mean.  The  probability  of 
any  other  value  of  x,  as  x  -\-  dx,  will  be 


but  A*=  ms\  P'  = 
\A\  being  the  sum  of  the  errors,  =  o. 
PSB***  -*»*-**•*•, 

since  d  =  o,  when  x  =  x^ 


P\P'\\ 
P\P'  :: 


dividing  the  second  ratio  by  the  third  term. 
If  m=i,    P:  P'  ::  i  :*-*« 


I2O  GEODETIC  OPERATIONS. 

In  this  single  observation  the  probability  of  zero-error,  as  in 
the  arithmetical  mean,  is  to  the  probability  of  error,  d,  as 
I  :  e  -AV2. 

As  h  is  the  measure  of  precision  of  a  single  observation,  /i* 
will  be  the  square  of  this  measure. 

In  the  expression  for  the  error  of  the  arithmetical  mean  we 
find  for  the  square  of  the  measure  of  precision  of  m  observa- 
tions K'm ;  therefore  the  measure  of  precision  of  the  arithmetical 
mean  of  m  quantities  is  h  \rm.  That  is,  the  measure  of  pre- 
cision of  the  mean  increases  as  the  square  root  of  the  num- 
ber of  observations. 


r.  =  — =-,        r  -  0.67456,        r.  =  .6745*. ; 
\  m 


therefore  £0  =  — — . 

Vm 


If  v,  i\,  z/a  .  .  .  be  the  observed  values  of  a  quantity  whose 
mean  is  x,  the  residuals  will  be  u,  u/t  «„,  or  x  —  v,  x  —  v 
x  —  vn  .  .  .     If  x  were  the  correct  value,  x  —  v,  x  —  vt,  x—  vu 
would  be  equal  to  A,  A' ,  A"  .  .  . 

and  m?  =  [//']  =  [«'] ; 


However,  this  does  not  consider  the  mean  errors  of  the  resid- 
uals.     Suppose  A  —  u  —  d,  A'  =  «'  —  d,  A"  =  u"  —  d,  .  .  . 


THEORY  OF  LEAST  SQUARES. 

s]  =  me  =  (u  —  <Tf  -f  («'  -  </)'  4-  (»"  - 

=  [«*]  _  2[«]^  +  *«</' 

=  M  +  w^*»         since  [«]  =  o. 


121 


*  may  be  taken  =  £„'  =  — 


so  that 


transposing 


r  =  0.6745 


,.=  ±  0.6745 


To  determine  the  probable  error  of  the  arithmetical  mean, 
we  find  the  difference  between  each  individual  result  and  the 
mean,  square  these  quantities,  and  divide  their  sum  by 


122 


GEODETIC  OPERATIONS. 


m(in  _  i)f  where  m  represents  the  number  of  individual 
results.  Extract  the  square  root  of  this  quotient,  and  mul- 
tiply by  0.6745 ;  the  product  will  be  the  probable  error  of  the 
arithmetical  mean. 

The  whole  operation  can  be  performed  by  logarithms. 

To  illustrate : 


Angle. 

u. 

u1. 

66°  -  54'  ~ 

12.  "5 
13.  5 
ii  •   3 

4-  i.i 

+  0.1 

+  2.3 

I.  21 
.OI 
5-29 

16.   5 

—  2.9 

8.4I 

12.   3 

+  1-3 

1.69 

IS-   5 

-  i-9 

3-6i 

Average 

13-  6 

20.22  =  [«s] 

The  probable  error  of  a  single  determination, 


The  probable  error  of  the  arithmetical  mean, 
'20.22 


;=  ±  0.6745 


=  ±  o".54- 


If  the  probable  errors  of  the  means  of  different  sets  of  deter- 
minations have  been  found,  their  relative  weights  may  be 
readily  ascertained.  Let  h,  /*„  A,  ...  be  the  measures  of  pre- 
cision, and  r,  rv  r,  .  .  .  the  probable  errors. 

Suppose  we  compare  our  individual  observations  with  a  fic- 
titious standard  whose  mean  error  is  e, ;  and  the  actual  observa- 


THEORY  OF  LEAST  SQUARES.  123 

tions  with  a  mean  error  s,  need  w  in  number  to  reduce  the  mean 

error  of  their  arithmetical  mean  to  £„ ;  this  gives  £„  —  —7=^,  or 

\  w 

Wf0  =  f,s.  Likewise  any  other  set  would  give  Wj£0/3  =  £,*,  or 
w^'*  =  TV?  ;  that  is,  wl  :  w  : :  f02  :  £0/a.  We  call  w,  w,  .  .  .  , 
etc.,  the  weights ;  they  are  reciprocally  proportional  to  their 
probable  errors. 

The  arithmetical  mean  of  n1  observations  of  weight  w,  «a  of 
weight  wv  etc.,  would  be 


_ 


\nw] 
or      x  =  -F — =;-,         and 


where  f1  is  the  mean  error  of  unit  weight. 

Let     v,  =  «!  —  x,        v^  —  7/2  —  x,        vt  —  n3  —  x, 

£/3  =  v*  -)-  f02  ;        but        W£"  =  £*  =  w,V^  +  W^*, 

and  £*  =  wj>?+  iv^*,  etc. 

If  m  be  the  sum  of  such  terms, 


124  GEODETIC  OPERATIONS. 

this,  substituted  in  the  value  of  £Q,  gives 


r_\wv^___ 

F«  -  V  \w\(m  -  i) 


In  figure-  or  station-adjustment,  if  the  number  of  repetitions, 
or  some  other  well-established  reason,  does  not  afford  weights 
for  the  averages  used,  the  reciprocals  of  their  probable  errors 
can  be  used.  While  in  the  development  of  the  foregoing 
formulae  there  were  a  number  of  assumptions,  and  some  ap- 
proximations to  cause  cautious  persons  to  distrust  the  absolute 
rigor  of  the  results,  it  will  be  apparent  to  all  that  the  arithmet- 
ical mean  deserves  a  confidence  that  varies  with  different  cases. 

Suppose  in  measuring  an  angle  ten  results  are  obtained  in- 
dividually differing  considerably  among  themselves.  In  an- 
other measurement  of  the  same  angle  ten  other  results  are 
secured  with  a  very  small  range ;  now,  if  the  average  be  the 
same  in  these  two  cases,  the  latter  would  be  more  readily  ac- 
cepted, as  the  residuals  are  individually  smaller.  So  we  need 
some  exponent  of  confidence  that  is  a  function  of  the  residuals  ; 
and  if  our  accepted  value  of  the  probable  error  is  not  absolutely 
correct,  it  will  afford  us  some  information  as  to  the  agreement 
of  the  individual  results  with  the  arithmetical  mean,  and  in  a 
number  of  different  determinations  it  gives  us  all  the  relative 
information  we  need. 

I  shall  add  just  here,  without  demonstration,  other  formulae 
in  general  use  in  determining  probable  errors: 

Probable  error  of  a  single  observation, 


THEORY  OF  LEAST  SQUARES.  125 

Probable  error  of  the  arithmetical  mean, 
r.  =  0.6745, 


If  m  =  number  of  observed  angles ; 

r  =  number  of  conditions  in  a  chain ; 


probable  error  of  an  adjusted  angle  =  /*  /—  -  times  prob- 
able error  of  an  observed  angle,  supposing  the  weights  ap- 
proximately equal  (Walker). 

If  an  angle  be  determined  by  a  direction-instrument,  its  value 
will  be  the  difference  of  two  directions ;  so  that  if  a  is  the 
probable  error  of  a  direction,  a  V 2  will  be  the  probable  error 
of  the  angle. 

If  rJ9  rv  r3  .  .  .  be  the  probable  errors  of  different  segments 
of  a  base-line,  the  probable  error  of  the  line  as  a  whole,  R  = 
Vr?  +  r;  +  r?... 

We  have  now  shown  how  to  determine  the  probable  error 
of  an  angle  or  a  base-line.  The  next  subject  to  consider  is  to 
what  extent  these  errors  in  an  angle  or  in  a  base  will  affect  the 
computed  parts.  As  the  errors  just  referred  to  are  small  in 
comparison  with  the  magnitudes  themselves,  we  may  omit  in 
all  the  discussions  into  which  they  enter  all  products,  and  pow- 
ers above  the  first.  All  geodetic  computations  are  based  upon 
formulae  relating  to  triangles,  so  we  will  investigate  those  ex- 
pressions which  are  of  most  frequent  occurrence.  Denoting 
the  sides  of  a  plane  triangle  by  a,  b,  and  c,  the  corresponding 
opposite  angles  by  A,  B,  and  C,  and  the  errors  with  which  they 
may  be  affected  by  da,  db,  dc,  dA,  dB,  and  dC,  we  can  find  by 


126  GEODETIC  OPERATIONS. 

computation  the  value  of  any  three  if  the  values  of  the  other 
three  be  known  (provided  one  be  a  side).  The  following 
formulae  are  those  most  frequently  used  : 

(1)  a.  sin.  B  =  &  .s'm  A  ; 

(2)  c  =  a .  cos  B  -f-  b .  cos  A  ; 

(3)  A+B+C=iSo°. 

As  da  represents  the  correction  to  a,  a-\-da  will  be  the  cor- 
rect value  of  a,  or  a  -f-  da,  b  -J-  db,  c  -|-  dc,  A  -\-  dA,  B-\-dB, 
•C -\-dCj  will  be  the  true  values.  Substituting  these  inequa- 
tions (i),  (2),  (3),  we  shall  have  : 


(4)  (a  +  da)  sin  (B  -f  dB)  =  (6  +  db}  sin  (A 

(5)  (c  +  dc)  =  (a  +  da)  cos  (B  +  dB)+(b  +  db)  cos  (A  +  dA) ; 

(6)  A+dA+B-\-dB+C+dC=  180°. 

But  sin  (B  +  dB)  =  smB-\-dB.  cos  B, 

since  sin  dB  =  dB,        and         cos  dB  =  i  ; 

also  sin  (A  -|-  dA)  =  sin  A  -j-  dA  cos  A  ; 

cos  (B  -f  <£B)  =  cos  B  —  dB  sin  B ; 
cos  (^4  +  dA)  =  cos  A  —  dA  sin  A 


Introducing  these  values  in  (4),  (5),  (6),  and  omitting  all 
products  of  da .  db,  etc.,  we  shall  get 


THEORY  OF  LEAST  SQUARES.  12? 

a  .  sin  B  +  da  .  sin  B  +  a  .  dB  cos  B  —  b  .  sin  A  -\-  db  .  sin  A 

-f-  b  .  dA  cos  A  ; 

c  -f-  afc  =  a  .  cos  B  -\-da.  cos  5  —  #  .  <s£Z?  .  sin  2?  +  b  .  cos  ^4 

.  cos  A  —  b  .  dk4  .sin  ^4; 


4-  £_{_  c+  dA  +  </£  -f  dC=  1  80°. 


Subtracting  equations  (i),  (2),  (3),  from  these  just  given,  we 
obtain: 


(7)  da  .  sin  B  +  0  .  dB  cos  5  =  db  .  sin  ^4  +  #  •  ^  cos  ^  ; 

(8)  «fc  =  ^  .  cos  B  —  a  .  dB  sin  B  +  db  .  cos  ^4  —  £  .  dL4  sin 

(9)  dA+dB  +  dC=  o. 


In  a  similar  way,  expressions  can  be  obtained  for  the  other 
parts,  as  : 


(10)     dc  .  sin  A  -f-  c  .  dA  cos  A  =  <afo.  sin  C-f-rt;  .  */£*.  cos  f; 
(i  i)    <#  .  sin  C  -f  *  .  dC  .  cos  C  =  dc  .  sin  5  +  c  .  dB  .  cos  B  ; 

(12)  <£z  =  dc.CQsB  —  c.dBsinB  +  db.coz  C—b.dCsin.  C; 

(13)  d#  =  da  .  cos  C  —  a  .  dC  sin  C-\-  dc  .  cos  A  —  c  .dA  sin  A. 


Suppose  that  in  a  triangle  c,  A,  and  B  are  given,  and  by 
means  of  them  the  values  a,  b,  and  C  are  computed.  Know- 
ing the  limits  of  errors  with  which  these  quantities  are  affected, 
it  is  required  to  find  the  limits  of  the  errors  with  which  the 
computed  quantities  are  affected  ;  that  is,  knowing  dc,  dA,  and 


128  GEODETIC  OPERATIONS. 

dB,  we  are  to  determine  da,  db,  and  dC.     From  equations  (7) 
and  (8)  we  have  : 


(14)  dC=-dA-dB-, 

(i  5)  da  .  sin  B  —  db  .  sin  A  =  —  a  .  dB  cos  B  -f  b  .  dA  cos  A  ; 

(16)  da  .  cos  B  +  <#  .  cos  ^  =  afc  -f  a  .  dBsin  B  +  b  .  dA  sin  .4. 

Multiplying  (15)  by  cos  A,  (16)  by  sin  A  and  adding,  we  get: 

(17)  da(s'm  A  .  cos  B  -f-  cos  ,/4  .  sin  B)  =  dc  .  sin  .4  -f-  £  .  dA 

—  a  .d£(cos  A  cos  B  —  sin  ^  sin  B}, 


or,  <fe  .  sin  (A  +  -5)  =  dc  .  sin  ^4  -f  b  .  dA  —  a  .  dB  .  cos  (A 


" 


sin 


Also  multiplying  (15)  by  cos  B,  (16)  by  sin  B,  and  subtracting, 
we  get  : 

db  .  sin  (A  +  .#)  =  d&r  .  sin  B  +  ^  .  dB  —  b  .  dA  .  cos  (<4  +  B}, 


-V  j_  > 

~     ^ 


_ 
sin  (^  +  ^)       sin  (A+B)~         sin  (^  +  B)' 

Since  sin  (^4  -J-  -5)  =  sin  C,  and  cos  (^4  +  ^)  =  —  cos  C,  we  can 
write  : 


.  smA       6.dA   .         7r>cos  C 

da  =  dc- — -^  -\-  -. — -^  -4-  a  .  dB- — ^, 

sin  C  '   sin  C    '  sin  C 


THEORY  OF  LEAST  SQUARES.  12$ 

,smB      a.dB 

db  —  dc- 


sin  C  '   sin  C    '  sin  C 

And,  again,  since  sin  A  :  sin  C\\a  :  c,  sin  B  :  sin  C  \\  b  :  c,  and 
for  cos  C :  sin  C  we  may  put  cot  C,  the  equations  then  reduce 
to  the  following  very  simple  form  : 

da  =  dca-  -f  -r1-^  -\-a.dB cot  C, 
<:     '    sin  (7 

db  =  dc .  — f-    .'   ^  +  b .  dA  cot  C ; 
c        sin  6 

or,  obtaining  the  relation  between  da  and  a,  db  and  £, 
da       dc    .     b  .dA 


a        c        <*  .  sn 

db       dc         a.  dB 


.  cot  C, 


.  cot 


Suppose  ^r  =  564.8,  ^4  =  61°  12'  12",  B  =  74°  16'  30",  and 
that  the  error  in  c  referred  to  c,  or  — ,  be  less  than  o.oooi,  and 
the  maximum  error  in  A  and  B  is  i". 

It  is  required  to  compute  — ,  -j-,  and  dC. 

log  b  =  2.8894998  log  a  =  2.8487359 

log  dA  =  4.6855749  log  sin  C  =  9.8458288 

7.5750747  2.6945647 

2.6945647 

4.8805100    =  log  of  second  term. 


130 


GEODETIC  OPERATIONS. 


log  dB  —  4.6855749 
log  cot  C  =  0.0072518 

4.6928267    =  log  of  third  term. 


First  term 

Second  term 

Third  term 


log  a 
logdB 


log  cot  C 
log  dA 


First  term 

Second  term 

Third  term 


o.oooiooo 
0.0000076 
0.0000049 

0.0001125    =  error  of  a  proportional  to  the 

side  a. 

2.8487359  log  b  =  2.8894978 

4.6855749  log  sin  C  =  9.8458288 

7.5343108  2.7353266 

2.7353266 

4.7989842    =  log  of  second  term. 

0.0072518 
4.6855749 
4.6928267  =  log  of  third  term. 

o.oooioooo 
0.00000629 
0.00000493 

O.OOOIH22  =  error  of  b  proportional  to  b. 


The  discussion  of  these  formulae  will  develop  some  very  in- 
teresting facts  concerning  the  best-shaped  triangles  to  make 
use  of  in  prosecuting  accurate  geodetic  work.  Upon  inspect- 
ing the  equations  it  will  be  seen  that  the  denominators  of  each 
term  of  the  second  members  is  sin  (A  -f  B},  consequently 
when  A  -\- B  is  nearly  180°,  or  when  C  is  very  small,  those 
terms  involving  sin  C  or  sin  (A  -f-  B)  as  a  denominator  will  be 
made  quite  large,  and  will  give  to  da  or  db  a  value  unduly  great. 


THEORY  OF  LEAST  SQUARES.  131 

Again,  the  second  members  will  have  the  smallest  value 
when  sin  C  has  its  greatest  value  or  when  C  =  90°;  supposing 
that  C  =  90°,  then  placing  sin  C  =  sin  90°  =  I,  the  equations 
reduce  to  the  form 


Should  dA  and  */2?  be  of  about  the  same  value,  and  b  be 
greater  than  a,  or  b  :  a  greater  than  I,  and  a  :  b  be  less  than  I, 
we  will  have  da  :  a  greater  than  db\b\  or  if  b  is  less  than  a 
we  will  have  da  :  a  less  than  db  :  b. 

From  which  we  can  see  that  it  will  be  best  when  a  =  b, 
consequently  when  A  =  B.  Remembering  what  has  just  been 
said,  we  see  that  the  right  isosceles  triangle  is  theoretically  the 
best  form  of  triangle  to  make  use  of. 

From  a  similar  discussion  it  will  be  apparent  that  if  b  or  a 
were  the  given  side,  the  smallest  error  in  the  other  quantities 
would  occur  when  B  or  A  =  90°. 

As  all  the  angles  cannot  be  each  equal  to  90°,  the  best  tri- 
angle is  the  equilateral.  A  similar  value  can  be  obtained  by 
direct  differentiation. 


sin  A    ,  ism  A     \        Js\nA\    ,         ,,smA 

—^.b,        da  =  d(- — £  .  b]  =  d(- — 5J .  b  +  db— 
sin  B  \sm  B     I         \sm  B'  sir 


4* 

\S1 


sin  £' 
fr.cosA  .s\nB .dA—b  .s\nA  cos£ .d 


sin  BJ    '  sin3  B 

It.cosA.dA        #.sin^     cos  B 
sin  B  sin  B      sin  B 


.dB. 


132  GEODETIC  OPERATIONS. 

b  c  b.cosA,dA 

Since  — — „  =  — — j.  we  may  write  for : — ^ — , 

sin  B      sm  A'  y  sin  B 


a  .  cos  A  .  dA 

: — =  a  cot  A  .  dA. 

sin  A 


&.s'mA          cos  B 
Also,  smce  a  =  --,  and  =  cot  B, 


b  sin  A    cos  B 

~~D~  -  ~ — 5  • dB  =  —  a .  cot  B .  dB ; 
smB      sm  B 


therefore,  da  —     —^db  —  a .  cot  B .  dB  -f  a  cot  A  .  dA. 
sm  r> 


Or,  by  logarithms, 

log  a  =  log  sin  A  -f  log  b  —  log  sin  B ; 
differentiating, 

da  _  cos  A  .  dA       db      cos  B  .  dB 
a  sin  A  b  sin  ^ 

=  cot  A  .  <W  +  ~  -  cot  £ .  dB ; 

<fc  =  0  cot  A  .  <£4  -f  ^  .  <#  —  «  cot  B  .dB ; 

d;  _  sin  A 

b  ~  sin  B  ' 


THEORY  OF  LEAST  SQUARES.  133 

hence,       da  =  —  —  ^db  —  a  cot  B  .  dB  4-  a  cot  A  .  dA. 
sin  /> 

Given  in  a  triangle  the  values  of  c,  A,  and  C  ;  required  to 
compute  the  values  of  «,  #,  and  B,  and  so  find  the  limits  of 
errors  of  the  latter,  supposing  the  errors  of  the  former  are 
known. 

From  the  equations  already  given  we  find  : 

dB=-dA  -dC\ 

da  .  sin  (A  +  ff)  =  dc  .  sin  A  -f  b  .  dA  —  a  .  dB  .  cos  (A  -\-  B)  ; 
db  .  sin  (^4  +  B)  -  dc  .  sin  B  -j-  a  .  </£-  b.dA.  cos  (4  +  £). 

But  sin  (A-\-  B)  =  sin  C,         and         cos  (A  +  £)  =  —  cos  C; 

hence,     da  .  sin  C  =  dc  .  sin  ^  -(-  £  .  dW  -j-  #  .  dB  .  cos  £7  ; 
db.smC=dc.smB  -\-a-dB  -{-b  .dA  .  cos  £7; 

dfc  .  a      b  .  dA       a  .  dB  .  cos  C 
da  =  ---  —  :  —  -^  H  --  :  —  :*  —  . 
c          sin  6  sin  C 

Since  dB  —  —  dA  —  dC,  we  may  write  : 

dC  .  cos  C" 


=  --  \-  -  —        ' 
c  a  .  sin  6 

since  b  —  a  cos  C  -\-  c  cos  A,  we  may  put  i:  cos  A  for  #  —  a  cos  £7, 

<&z       ^   ,    c  .  cos  ^(  .  dA  - 

then  -  —  ---  ----  .    r      —  dC  .  cot  C  ; 

#         *    '         «  .  sin  6 

but  «  .  sin  C  =  c  .  sin  A, 

therefore          ™  =  --  +  dW  .  cot  ^4  -  dC  .  cot  67. 


134  GEODETIC  OPERATIONS. 

db      dc.s\r\B  t    a.dB       dA.  cos  C 
Likewise,         —,-  =  —j—.  —  7?-  -I-  -r-.  —  -^-\  --  :  —  7=;  —  , 
b         bsmC        b  sin  C          sine 

_  dc       a  .dA        a.dC       dA  cos  C 
~  c        b  sin  C      b  sin  C         sin  C   ' 

&  cos  C  ~  adA        a'dC 


~~c  tr.s'mC         ~  b  sin  C 

_dc      c  .  cos  B  .  dA         a  .  dC 
~  c  b  sin  C  b  .  sin  C' 

dc  a.dC 

=  ---  dA  .  cot  B  —  -r—^—^- 

c  b  .  sin  C 

Let  c  =  450,  A  -  53°  19'  16",  C  =  61°  42'  32",  we  will  find 
by  computation  ^  =  64°  -58'  12",  a  =  409.855,  b  =  463.05. 

Suppose  that  c  be  reliable  to  within  o.oooi  of  its  entire 
length,  so  that  dc  -v-  c  =  o.oooi,  and  let  dA  =  dC  =  5"  : 

log  dA  =  5.3845449  log  dC=  5.3845449 

log  cot  A  =  9.8720420  log  cot  C  =  9.7309796 

dA  .  cot  A  =  0.00001805 
dc  -T-  £  =  o.oooioooo 
*#7  .  cot  C  =  o.oocoi  3047 


da-^a  —  0.000131097 
log  *W  =  5.3845449 
log  cot  B  =  9.6692660 

log  dA.  cot  ^  =  5.0538109  =  0.000011319 

log  a  =  2.6126301 
log  dC  =  5.3845449 


=  7.9971750 
log  £  =  2.6656276 
log  sin  C  =  9.9447545 

log  (J.sin  C)  =  2.6103821,  log  (a  .  dC  -5-  £  .  sin  C)  =  5.3867929 


THEORY  OF  LEAST  SQUARES.  135 

dA  .  cot  B  =  o.ooooi  1319 
a .  dC  -7-  b  .  sin  C  =  0.000024366 

Subtracting  the  sum  of  these  two  quantities  from  dc  -f-  c, 
we  get 

db  -r-  *  =  0.000064315,        dB  =  —  ($"  -+-  5")  —  —  10". 

Since  */£"  may  be  either  positive  or  negative,  we  may  select 
the  sign  which  will  give  the  maximum  value  for  the  error 
da  -~  a.  We  have  therefore  added  the  expression  dC .  cot  C. 

In  the  following  discussion,  we  will  do  the  same,  suppos- 
ing the  errors  committed  in  measuring  A  and  B  were  nearly 
equal  and  of  the  same  sign. 

When  they  are  nearly  equal  and  of  the  same  sign,  according 
to  the  equation  already  given,  they  will  compensate  one 
another.  But  should  dc  have  the  same  sign  as  dA  and  dC,  it 
would  lessen  the  value  of  da  when  we  have  dC .  cot  C  greater 
than  dA  .  cot  A,  since  the  amount  of  error  committed  in  measur- 
ing the  angles  is  less  than  that  of  the  measured  side. 

The  value,  therefore,  of  da  -f-  a  will  become  the  least  when 
C  is  less  than  A  and  acute.  But  if  A  be  obtuse,  consequently 
cot  A  negative  (assuming  dA  and  dC  to  be  positive),  da  -r-  a 
will  become  the  least  when  C  is  acute,  for  then  the  last  two 
terms  are  to  be  subtracted  from  dc  -=-  c. 

For  A  =  90°,  the  third  term  will  disappear  entirely, 
which  circumstance  will  be  advantageous  with  respect  to  da 
-f-  a. 

In  regard  to  the  side  a,  A  must,  therefore,  be  either  a  right 
angle  or  obtuse,  and  fas  small  as  possible. 

The  same  reasonings  apply  with  respect  to  b,  with  the  ad- 
ditional circumstance  that  B  should  be  also  very  small. 

Therefore,  in  the  present  instance,  a  large  value  for  A  and 
small  values  for  B  and  C  will  produce  the  least  errors. 


136  GEODETIC  OPERATIONS. 

Suppose  we  have  the  two  sides  and  the  included  angle  with 
their  limiting  errors  given,  and  wish  to  find  the  limiting  errors 
of  the  unknown  or  computed  parts  ;  that,  is  having  a,  b,  and  C, 
to  compute  the  value  of  the  errors  in  c,  A,  and  B. 

From  equations  (7),  (8),  and  (9)  we  have 


(1)  d 

(2)  a  .  dB  .  cos  B  —  b  .  dA  .  cos  A  —  db  .  sin  A  —  da  .  sin  B  ; 

(3)  dc  +  a  -  dB  .  sin  B  -f-  b  .  dA  .  sin  A  =  db  .  cos  A  +  da  .  cos  B. 

Substituting  for  dA,  —  (dC+  dB},  (2)  becomes 
(4)0  .  dB  .  cos  B  -f  b  .  cos  A  (dB  +  dC}--da  .  sin  B  +db  .  sin  A. 
By  expanding  and  transposing  we  get 

(5)  (a  .  cos  B  -f  b  .  cos  A)dB  =  —  da  .  sin  B  -f  db  .  sin  y4 

—  b.dC.cosA. 

Putting  c  for  #  .  cos  B  -\-  b  .  cos  ^4, 

(6)  <:  .  dB  =  —  da  .  sin  B  +  <#  .  sin  A  —  b  .  dC  '.  cos  A, 
likewise, 

(7)  c.dA  =  dA  .  sin  B  —  db  .  sin  A  —  a  .  dC  .  cos  B. 
By  transposing  (3),  we  have 

(8)  dc  =  da.cosB  +  db.cosA—a.dB.s\r\B  —  b.dA.  sin  A. 


THEORY  OF  LEAST  SQUARES.  137 

Multiplying  this  through  by  c  and  substituting  torc.dAand 
c  .dB,  the  values  given  on  page  136,  we  get 

(9)  c  .  dc  —  c.  da  .  cos  B  -f-  c.  db  .  cos  A 

—  a  .  sin  B(—  da  .  sin  B  -j-  ^  .  sin  ^4  —b.dC.  cos  ^4) 

—  £  .  sin  ^4  (d0  .  sin  B  —  db  .  sin  A  —  a.dC  .  cos  Z?). 
=  c  .da.cosB  -{-c  .dBcosA 

-f-  («  .  sin  B  —  b  sin  .4  )(</<&  .  sin  B  —  db  sin  A) 
-}-  a.b.  dC(cos  A  .  sin  B  -f  sin  A  .  cos  ^). 

The  first  factor  =  o,  and  the  last  is  sin  (A  +  B)  or  sin  C; 
therefore 

(10)  c  .  dc  =  c  .  da  .  cos  B  -\-  c  .  db  .  cos  y2  +  a  .  #  .  d£\  sin  C. 
By  expansion  and  substitution,  (9)  becomes 

(10)    dc  =  da  .  cos  B  +  db  .  cos  ^4  -j-  #  .  £  .  dC  .  sin  C  -f-  ^  ; 
Dividing  (6)  and  (7)  by  c,  we  get 

(i  i)  dB  =  db  .  sin  A  -r-  c  —  da  .  sin  B  -f-  c  —  b  .  dC  '.  cos  ^4  -=-  £  ; 
(12)  dk4  =  —  db  .  sin  ^4  -=-  c  -j-  dk  .  sin  B  -i-  ^  —  a  .dC.  cosB  -^-c. 

In  computation  this  formula  is  used  like  those  already  illus- 
trated, so  it  will  be  needless  to  give  a  solution  of  an  example 
of  this  kind. 

The  only  remaining  case  is  when  we  have  the  three  sides 
with  their  limiting  errors  to  find  the  limiting  errors  of  the  com- 
puted angles.  The  discussion  of  this  problem  is  of  interest 
simply  from  a  theoretical  point  of  view,  since  such  a  case  will 
never  arise  in  any  one's  experience. 

Rewriting  equations  already  deduced  (page  1  27),  we  start  with 


(  i)     6.sinA.dA+a.smJ3.dl?  =  da.cosJ8  -\-db  .  cos  A  —dc  ; 

(2)  b.cosA.dA—a.  cos  B.dB  —  da.sinB  —  d 

(3)  dA  +  dB  +  dC=o. 


138  GEODETIC  OPERATIONS. 

Solving  the  first  two  equations  with  reference  to  dA  and  dB. 
and  substituting  in  the  results  the  values  of  dA  and  dB  in  (3), 
and  putting  sin  C  for  sin  (A  -j-  £),  —  cos  C  for  cos  (A  -f-  B\ 
we  will  obtain 

(4)  dA  =  da  -j-  b  .  sin  C  —  db  .  cos  C  -f-  #  .  sin  C  —  dfc  .  cos  j5 

-=-  £.  sin  £7; 

(5)  ^/.Z?  =  —  da  .  cos  £7  -j-  a  .  sin  £T  -j-  db  -~  a  ,  sin  C  —  dc  .  cos  A 

-f-  tf  .  sin  (7; 

(6)  dC=da(b  .  cosC  -  a)+  a  .  &  .sm  C  +  db(a  .cosC-  b) 

-r-a.  &.sinC-{-  dc(a  .  cos  .Z?  +  £  .  cos  A)  •—  a  .  b  .  sin  C. 

Should  we  have  da  -~  a  =  db  -h  b  =  dc  -j-  c,  the  errors  of  the 
sides  would  be  proportional  to  the  sides  themselves. 

The  defective  triangle  would  then  be  similar  to  the  true 
triangle,  and  the  corresponding  angles  would  be  equal  each  to 
each,  and  we  would  have  dA  =  dB  =  dC  '=  o. 

When  the  three  angles  of  a  triangle  are  observed,  the  differ- 
ence between  their  sum,  after  subtracting  the  spherical  excess, 
and  180°  is  the  total  error  of  the  triangle. 

Let  us  call  this  E,  the  errors  of  the  individual  angles  x,  y, 
and  z,  with  respective  weights,  «,  v,  w, 


By  preceding  theory  ux*  -\-  vy*  +  wz*  =  a  minimum  ;  differ- 
entiating with  respect  to  x,  y,  and  2,  we  get  ux  =  vy  =  tvz. 


u  w  u  w 

wvy  -\-  wuy  -\-  uvy  =  wuE ; 
_  wuE  wvE  uvE 

~  wv -\-wu-\-  uv'  ~  wv  -f-  wu  -\-  uv1  Z  ~  wv-\-wu-\-uv 


THEORY  OF  LEAST  SQUARES.  139 

The  limits  of  errors  may  be  found  in  a  similar  manner  for 
all  combinations  of  triangles;  hence  a  polygon  may  be  decom- 
posed into  triangles  and  the  limits  of  error  found  by  the 
methods  just  described. 

This  method  is  not  altogether  satisfactory,  since  in  the  com- 
putation of  the  error  in  each  triangle  we  use  the  errors  of  only 
two  of  the  angles,  ignoring  the  third. 


TT  sin  A    A 

From  trigonometry,  we  have  a  =  -. — .  b\ 


by  differentiation,  we  have 

da=acotA.dA—acotB.dB,    .    .     .     .     (i) 

b  being  a  constant. 

Let  of,  fi,  and  y  be  the  measured  angles,  and  A,  B,  and 
C  the  correct  values.  The  triangle  error,  after  correcting  for 
spherical  excess,  is  a-\-  ft  -j-  y  —  180°,  one  third  of  which  may 

be  attributed  to  each  angle,  so  the  error  in  a  =  —  , 

and  the  correct  value 


also,   B  =  p--  =  -,    (3) 


iSo0  _2y-/3-a  +  180° 
C-y-~         —  —        -.      (4) 


As  A  and  B  depend  upon  a,  fi,  and  y,  the  total  error  in  the 
side  a,  or  ea  will  be  a  function  of  a,  ytf,  and  y; 


140  GEODETIC  OPERATIONS. 


In  which  fa)  f/s,  and  fy  are  the  probable  errors  in  a,  ft,  and  y, 
and  in  (i)        da  =  aco\.A.dA — a.cotB.dB., 

so  we  must  obtain  dA  from  (2),  in  terms  of  ex,  ft,  and  y. 
Likewise,  dB  from  (3),  or  we  may  write  (i) 

I2a  —  13  —  v+  i8o°\ 
da  =  a  cot  A  .  d\ — ! J 

,(2/3  —  a  —  y-\-  i8o°\ 
—  acotB.d{— -  — J.  ...     (6) 

Differentiating  (6)  with  respect  to  a,  ft,  and  y,  we  have 
da       2 


(7) 


^ 


^  --  a  cot  A  —  -a.cot£—a\—  -  cot  A  --  Cot2?]|(8) 
-f  t 


Squaring  (7),  (8),  and  (9), 


=  ia'  cot8  ^  +  *-(?  cot  ^  .  cot  B  +  -#'  cot5  B  ; 


=-a*  cot9  ^  +   a  cot  ^  .  cot  B  +  z2  cot2  B  ; 
-a*  cot2  ^4  -  --«2  cot  A  .  cot  ^  +  -a2  cot2  .5  ; 


THEORY  OF  LEAST  SQUARES.  141 

therefore,  supposing  £  a  =  fp  =  fy  =  e,  we  have 
(da  V  ,    (da  V 


=  €'[fa'  cot2  A  4-  ftf2  cot  A  .  cot  B  +  f  fl9  cot5  E\ 
=  f  «V(cot'  ^4  +  cot  ^  .  cot  £  +  cot2  B)  ; 


=  fa  1/|(cota  ^4  -f  cot  A  .  cot  .5  +  cot2  E). 


As  £a  is  small,  the  second  member  can  be  converted  into  a 
linear  unit  by  writing  it  equal  to  its  algebraic  value  times  sin 

l". 


ea  =  ea.  sin  i"  i/|(cot2  ^  4-  cot  A  .  cot  .#  +  cot2  B).    (10) 

This  is  a  rigorous  expression  for  the  probable  error  of  a  side, 
as  computed  from  a  base  supposed  to  be  free  from  error. 
The  side  a  of  the  first  triangle  may  be  regarded  as  the  accu- 
rate value  of  the  base  of  the  next  triangle,  and  the  probable 
error  of  another  side  computed,  and  so  on  through  the  entire 
chain.  So  we  may  put  for  („,  the  error  of  the  last  side, 


£„  =  easin  i"  V |^(cota  A  +  cot^  .  cot  B  +  cot2  B).    (11) 

In  which  2  represents  the  sum.  In  determining  the  angles  to 
be  used  in  this  formula,  it  must  be  remembered  that  A  is  op- 
posite the  side  whose  error  is  being  -determined,  and  B  is  op- 
posite the  side  whose  error  was  last  computed. 


142  GEODETIC  OPERATIONS. 

To  illustrate: 

Starting  with  the  base  b,  we  first  pass  to  u ; 

in  this  case          A  is  I,  B,  3  ; 

then  to  x,  A  is  4,  B,  5  ; 

then  to  #,  A  is  7,  ^,  8. 

Hence,     2  cot'  ^  =  cot2  (i)  +  cot2  (4)  -f  cot2(;)  . . .  et 

In  (n),  £  is  the  average  angle-error  in  the  chain.  If  the  proba- 
ble error  of  the  base  be  f/,  this  error  will  be  carried  through 
the  chain  without  augmentation  or  diminution,  owing  to  inac- 
curacies in  the  angles,  but  it  will  be  increased  in  the  ratio  of 
the  length  of  the  computed  line  to  the  base.  In  the  first 

& 

computed  side  a,  the  error  from  this  source  will  be  —  £&'.  Sup- 
pose this  be  £/,  then  in  the  next  triangle,  if  c  is  computed  from 

c  c     a  c     , 

a,  the  error  ey  =  —  .  sx   =  —  .  j- .  e6    =  Te6  ,  and  so  on  through 
a  a     b  b 

the  entire  chain;  so  if  n  be  the  last  line,  £„'  =  r •  ftf-     The  total 

o 

error,  £,  from  both  sources,  will  be  E  =  Vfn'3  -f-  £„".  If  each 
side  of  each  triangle  has  been  computed  by  two  different 
routes,  the  value  for  E  must  be  divided  by  V^2,  or,  E  = 


CALCULATION  OF   THE    TRIANGULATION.  143 


CHAPTER  VI. 

CALCULATION  OF  THE  TRIANGULATION. 

HAVING  assumed  in  the  field  that  we  had  a  line  of  known 
or  approximately  known  length  for  a  base-line,  we  measured 
the  angles  of  all  the  triangles  of  our  net  a  sufficient  number  of 
times  to  eliminate  instrumental  errors  ;  and  now  wish  to  com- 
pute the  distances  of  all  the  stations  from  one  another,  as  far 
as  possible. 

When  the  three  angles  of  a  triangle  are  measured  with  the 
same  care,  it  will  be  found  that  their  sum  will  not  equal  180° 
-|-  spherical  excess,  and  when  two  individual  angles  are  meas- 
ured separately  and  then  as  a  whole,  the  sum  of  the  two  will 
not  equal  the  two  when  treated  as  a  single  angle;  and,  again, 
the  sum  of  the  angles  that  complete  the  horizon  will  always 
differ  from  360°. 

The  problem  then  is  to  find  results  from  a  number  of  ob- 
served values  that  will  approach  the  nearest  to  the  truth,  and 
at  the  same  time  eliminate  those  discrepancies  just  referred  to. 

There  are  two  classes  of  conditions  that  should  be  fulfilled : 

(a)  the  sum  of  the  individual  angles  should  equal  the  meas- 

ured whole  ; 

(b)  the   sum  of  all   the  angles  completing  the  horizon  should 

equal  360°. 

The  operation  of  filling  these  conditions  is  called  station-ad- 
justment. 

(c)  The  three  angles  of  a  triangle  should  equal  180°  ; 

(d)  The  length  of  every  side  should  be  the  same,  regardless  of 

the  route  by  which  it  is  computed. 


144 


GEODETIC  OPERATIONS, 


The  filling  of  these  conditions  is  called  figure-adjustment. 
These  two  adjustments  are  to  be  effected  simultaneously,  since 
the  same  quantities  occur  in  each. 

The  method  of  adjustment  will  depend  on  the  way  in  which 
the  angles  are  read ;  whether  with  a  repeating-theodolite,  or 
direction-instrument.  If  with  the  former,  the  average  value 
obtained  for  each  angle  will  be  the  quantity  that  enters  the 
equations  formed  by  the  expressed  conditions. 

Before  writing  these  equations  we  must  correct  the  angles 
for  run  of  micrometers,  as  already  explained  on  page  35,  and 
for  phase. 

The  latter  is  the  effect  of  sighting  to  the  illuminated  por- 
tion of  a  signal  instead  of  the  centre  ;  it  is  only  appreciable 
when  a  tin  cone,  or  some  large  reflecting  surface,  is  observed 
on.  This  bright  spot  will  be  exactly  in  the  line  to  the  centre 
when  the  sun  is  directly  behind  the  observer,  and  furthest 
from  the  centre  when  the  sun  is  at  an 
angle  of  90°  with  the  cone  and  ob- 
server. 

Let  £7be  the  centre  of  the  signal  and 
O  the  the  position  of  the  observer,  the 
distance  in  the  figure  being  greatly 
shortened,  proportioned  to  the  size  of 
the  signal. 

The  rays  of  the  sun  may  be  re- 
garded as  parallel  and  illuminating 
half  of  the  signal,  as  A  SB.  Of  this 
the  observer  sees  only^S^F;  this  he 
bisects,  sighting  to  D  instead  of  C. 
This  causes  an  error  equal  to  the 
angle  COD. 

Let  SCG  =  x,  EC  =  r,  OC  =  D, 

and  COD  =  d.     KF\s  the  projection  of  the  visible  arc,  and 
CD  =  \EK.     AK,  being  perpendicular  to  EF,  and   FAE  a 


CALCULATION  OF  THE    TRIANGULATION.  145 

right  triangle,  AK*  =  EK.KF  =  EK.  2EC  (nearly),  or, 


but        AK=EC.sinACE, 


-  2EC' 
hence,  AK*  =  EC* .  sin*  ACE, 

EC*,  sin* ACE      EC.  sin* ACE 


and       EK  = 


2EC 


As  ACE  Is  small,  we  can  write  sin*  ACE  =  4  sin*  %  ACE,  EK 
=  2r  .  sin*  $ACE,  but  ACE  =  GCS,  both  being  complements  of 

f*  fC 
ACG,  or,  ACE  =  x  ;  also  CD  =  —  -  ;  substituting,  CD  =  rsin* 


\x.     In  the  right  triangle  OCD,  sin0  =  -^  =  T  Sl^  **.     As  (9  is 
small,  sin  6  =  6.  sin  i",  or,  0  =      S          . 


This  correction  is  to  be  subtracted,  when  the  sun  is  to  the 
right  of  the  observer,  and  added  when  the  sun  is  to  the  left.  In 
the  case  of  independent  angles,  if  both  objects  observed  need  a 
correction  for  phase,  the  two  individual  corrections  are  to  be 
subtracted  if  they  have  opposite  signs,  and  added  when  they 
have  the  same  signs. 

In  the  principle  of  directions,  each  direction  should  be  cor- 
rected for  phase,  using  only  the  average  direction  in  applying 
the  correction,  and  at  all  times  measuring  the  angle  x  about 
the  mean  time  of  the  series  of  observations. 

A  similar  correction  is  to  be  applied  when  an  eccentric  sig- 


146  GEODETIC  OPERATIONS. 

nal  was  sighted ;  in  this  case  it  is  necessary  to  know  the  per- 
pendicular distance  from  the  centre  of  the  signal  to  the  line 
joining  the  observed  and  observing  stations.  This  will  form  a 

right  triangle  in  which  sin  6  =  -=-,  or  0  =  -= — : 77,  in  which 

D  Z/,  sini" 

6  =  correction  ;  r,  the  perpendicular,  and  D,  the  distance  be- 
tween the  stations.  This  correction  is  additive  when  the  point 
observed  is  within  the  angle  formed  by  the  centres  of  the  two 
stations,  and  subtractive  when  it  falls  without. 

This  is  but  a  special  case  of  reduction  to  centre,  discussed 
on  page  196,  and  like  the  latter  can  be  applied  later  as  well 
as  at  this  point. 

With  the  average  angles  corrected  we  wish  to  find  those 
values  that  will  fulfil  the  required  conditions  and  at  the  same 
contain  the  largest  element  of  truth.  We  have  seen  that  the 
arithmetical  mean  renders  the  sum  of  the  squares  of  the  resid- 
uals a  minimum,  and  that  the  most  probable  value  of  a  num- 
ber of  disagreeing  results  is  the  one  that  makes  the  squares  of 
the  errors  a  minimum.  So  we  shall  now  look  for  that  most 
probable  value  which  will  fulfil  the  conditions. 

Suppose  we  have  a  series  of  a  observations,  giving  m  for  the 
arithmetical  mean,  and  a  series  of  &,  giving  n  for  the  mean  ;  the 
relative  value  of  these  two  means  would  be  to  each  other  as 
a:  b. 

Consequently  the  larger  number  of  equally  good  observa- 
tions we  have,  the  better  relative  value  we  will  get  for  the 
mean.  If,  therefore,  the  first  arithmetical  mean,  v^  be  obtained 
from  a  series  of  at,  the  second,  z/a,  from  a^  .  .  .  the  «th,  vn,  from 
an,  we  will  have  for  the  most  probable  value  of  x, 


CALCULATION  OF   THE    TRIANGULATION. 
If  an  angle  be  measured 

10  times  with  the  result  18°  18'  12", 
8  times  with  the  result  18°  18'  2" ', 
5  times  with  the  result  18°  18'  21", 
4  times  with  the  result  18°  1 8'  30", 


10  +  8+ 


On  page  123  it  was  shown  that  the  residuals,  or  individual 
errors,  squared  were  multiplied  by  their  weights,  and  these 
products  summed,  to  give  the  square  of  the  probable  error  of 
the  observations  as  a  whole.  Then,  since  this  probable  error 
is  obtained  by  taking  that  value  which  reduces  the  residuals 
squared  to  a  minimum,  the  sum  of  the  individual  errors 
squared,  each  multiplied  by  its  respective  weight,  must  assume 
the  form  which  renders  it  a  minimum. 

By  way  of  illustration,  let  us  take  the  following  example  : 
Suppose  A,  B,  and  C,  be  three  angles  in  a  plane  around  a 
point  as  a  common  vertex,  and  amounting  to  360°.  Suppose 
the  measured  values  be  A,  B,  and  C\  at,  bt,  and  ct  their  true 
values,  and  a,  b,  and  c  the  errors  of  A,  B,  and  C,  so  that  we 
have  A  -\-a  =  a,B-\-b  =  b,  C  -\-  c  =  c/,  also,  a4  +  b,  +  c, 
=  360°. 

From  a  set  of  10  measurements  A  =  120°  15'  20"  ; 
From  a  set  of  12  measurements  B  =  132°  16'  30"  ; 
From  a  set  of  15  measurements  C  =  107°  28'  19". 


148  GEODETIC  OPERATIONS. 

=       360°  oo'  09" 
~        ^°°  oo/  °°" 


a  +  £  +  c  =  -    00°  oo'  09"  (i) 

Taking  the  sum  of  the  squares  of  these  errors,  #a  -j-  b*  +  £*, 
and  multiplying  each  by  its  respective  weight, 

I0rt'+  I2#'+  15*:*.  (2) 

From  (i)  c  =  —a  —b  —  9  ; 


£  -f-  iSa  +  1  8^  ; 

=  15(81  +  rt2  +  &  +  2^  +  1  8*  +  1 
=  1215  -f  150'+  15^+ 


substituting  this  value  for  15**  in  (2),  we  get 

270^+  1215.  (3) 


According  to  principles  already  explained,  we  obtain  the  dif- 
ferential coefficient  with  respect  to  a  and  b,  and  place  each  re- 
sult equal  to  zero  ; 

$oa  +  30^  +  270  =  o  ;         50  +  3^  +  27  =  o  ; 
270  =  0;        50  +  9^  +  45  =o; 


therefore,  b  +  3"  =  o,  b  =  —  3",  a  =  —  3".6;  substituting  in 

(l),<r=:-2".4. 

The  same  result  may  be  obtained  by  using  an  indeterminate 
coefficient,  and  afterwards  eliminating  it, 

29(a 


CALCULATION  OF   THE    TRIANGULATION.  149 

we  take  2cp  to  avoid  the  use  of  fractions.  Differentiating  this 
with  respect  to  a,  b,  and  c,  and  placing  the  results  equal  to 
zero,  we  get 

2oa  -f-  2<p  =  o        or        loa  -\-  cp  —  o ; 
24&-{-2<p=o  i2#-f-0»  =  o; 

30^  -\-  2cp  =  o  1 5^  +  ^  —  o. 

Eliminating  q>  from  two  of  these  equations,  we  get 

loa  —  I2b  =  o; 

I2b  —  i$c  =  o; 

also,  a-\-b-\-c-\-g  =  o. 

By  the  simple  elimination,  we  get 

«=-3".6,    *=-3"     ^=-2"4- 

A  =  120°  1 5' 20"  —  s".6  =  120°  15'  i6".4; 
B  =  132  16  30  —  3  .o  =  132  16  27  .o; 
C  =  107  28  19  —  2  .4  =  107  28  16  .6 ; 
aiJrj)iJrci  =  360°  oox  oo". 

The  above  is  the  simplest  case  in  practice ;  that  is,  when  only 
one  condition  is  to  be  fulfilled.  Let  us  pass  to  a  more  com- 
plicated case,  or  when  several  conditions  are  to  be  fulfilled. 

Suppose,  in  Fig.  13,  we  have  from  repeated  measurements 
the  following  results : 

(1)  MON=    68°  37'    i "  with  the  weight    5; 

(2)  MOP  =  140     2  19    with  the  weight  10  ; 

(3)  NOQ  =  134    15  41    with  the  weight  20  ; 


GEODETIC  OPERATIONS. 


(4) 
(5) 

(6) 

(7) 
(8) 


NOR  =  211  56  10  with  the  weight  15  ; 

FOR   =  140  30  40  with  the  weight  12  ; 

MOQ  =  202  52  46  with  the  weight  18  ; 

NOP  =    71  25  38  with  the  weight  16 ; 

QOR  =    77  40    6  with  the  weight  20. 


Upon  inspection  it  will  be  seen  that  the  following  conditions 
should  be  fulfilled: 


(I) 
(2) 

(3) 
(4) 


(2)  -(i)  =  (7): 
(4) -(3)  =  (8): 

(5) +  (7)  =  (4) 
(6)  -  (3)  =  (i). 


(A) 


Denoting  the  corrections  to  the  angles  by  a,  b,  c,  .  .  .  k, 

(1)  [(2)  +  J]  -  [(i)  +  a]  =  [(7)  +  £•]  ;  ^ 

(2)  [(4)  +  «/]  -  [(3)  +  c\  =  [(8)  +  *] ;  •  I 

(3)  [(5)  +  e]  +  [(7)  +£•]  =  [(4)  +  </] ;  f 

(4)  [(6)  +/]  -  [(3)  +  c\  =  l(i)  +  a].    J 


(B) 


CALCULATION  OF   THE    TRIANGULATION.  151 

Substituting  in  these  equations  the  angles  designated  by  (i), 
(2),  .  .  .  (8),  they  reduce  to 


(C) 


These  are  the  relations  that  must  exist  between  the  correc- 
tions that  are  to  be  applied  to  the  different  angles. 

Squaring  each  symbolic  correction  and  multiplying  each  by 
its  respective  weight,  we  have 


(I) 

b  —  a  —g=  — 

20" 

(2) 

d-c-h= 

23 

(3) 

e+g-d= 

8 

(4) 

f-c-a= 

4   • 

$0*  +  io£'  -f  20^'  +  i$d*  +  12*'  +  i8/3  +  i6g*  +  2oh\  (D) 
From  the  equations  at  (C)  we  obtain 


(i),  £  =  «+£•  -20;  (2),  c  =  d—  /*  —  23  ; 
(3),  '  =  <*- 


Substituting  these  in  (D), 


-  2o)3  +  2o(^/-  h  -  23)'  +  i5^z 
-  A  -  19)'+ 


must  be  a  minimum. 

The  square,  omitting  constants,  gives  : 


'  -f-  iotf2  +  ic^"2  +  2cwg-  —  400^  —  400^+20^"  4- 

—  920^-f920/2+  15^'+  I2^a+  I2^2  — 

-  192^-+  i8«2  +  i$d*  +  18^  -f- 

—  684^  —  684^  +  684/2  +  j6£-2 


152  GEODETIC  OPERATIONS. 

Differentiating  this  and  placing  the  coefficient  of  each  equal  to 
zero,  we  have 


330        i&   +  10^-  —  i<  =  542  ; 
180  -f-  6$d  —  I2g  —  38/2  =  706  ; 

$a  -    6d  +  19^  ='  148  ; 

ga  +  19^  —  297*  =  401. 

The  solution  of  these  equations  gives  a  =  6.4.6,  d  =  5.89,^  = 
7.95,  and  h  =  —  7-97,  which  values  substituted  in  C,  give  b  = 
—  5.59,  c  —  —  9-14,  e  =  5-94>  and  f  =  1.32.  Since  the 
errors  are  to  be  obliterated  in  applying  the  correction,  each 
correction  must  have  the  opposite  sign  to  its  error  ;  so  that  if 
the  above  represent  the  errors,  they  are  to  be  applied  with 
contrary  signs  to  the  respective  angles,  which  reduce  the 
angles  to  : 

(i)=    68°  36'  54".54  ; 

(2)  =  140      2  24  .59  ; 

(3)  =  134  15  50  .14; 

(4)  =  211  56    4  .11  ; 

(5)  =  140  30  34  .06  ; 

(6)  =  202  52  44  .68  ; 

(7)  =    71  25  30  .05  ; 

(8)  =    77  40  13  .97. 

H.K  £         In  order  to  furnish  practice, 

the  following  observed  angles 

B.K  >>^  \  /       are    taken    from    the  author's 

record-book.  The  corrected 
results  are  given,  so  that  those 
adjusting  them  can  verify  their 

C.T  ----  """  work.     This,  however,  can  be 

FlG-  I4  done  by  seeing    if  the  condi- 

tions are  fulfilled  when  the  corrected  values  are  taken.     The 
weights  are  equal,  and  so  can  be  omitted. 


CALCULATION  OF   THE    TRTANGULATION. 
Observed.  Corrected. 

(1)  CTto  BK=  36°24/23//.25  =  36°  24'  22". 73  ; 

(2)  CTto  HK=  49  53  49  .36  =  49  53  51  .61 ; 

(3)  CT  to  C     =  95  06  40  .80  =  95  06  39  .05  ; 

(4)  BKtoHK=  13  29  31  .11  =  13   29  28  .86; 

(5)  BKto  C     =  58  42  14  .55  =  58  42  16  .30. 


B.B 


C 


153 


(I) 

(2) 

(3) 
(4) 
(5) 
(6) 
(7) 
(8) 

Observed.                          Corrected. 

BB  to  C     =  26°  44'  50".  5  7  =  26°  44'  57".82  ; 
BB  to  /f    =  62  55  56  .14  =  62  55  47  .315 
BB  to  £AT  =  85  08  27  .43  =  85  08  29  .005 
C     to  H    =  36   10  41  .57  =  36   10  49  .495 
C     to  BK  =  58  23  31  .86  =  58  23  31  .185 
H    to  BK  =22   12  40  .79  =  22   12  41  .69; 
H    to  CT  =  53  09  ii  .98  =  53  09  10  .18; 
BK  to  CT  =  30  56  26  .69  =  30  56  28  .49. 

In  a  large  number  of  condition  equations  the  above  opera- 
tion may  be  considered  long  and  tedious,  so  that  one  of  the 
following  methods  may  be  found  preferable. 

Suppose  we  have,  as  the  result  of  the  same  number  of  related 
quantities  x,  y,  and  2,  the  values  N^  Nv  and  N3,  giving  the 
equations : 


154  GEODETIC  OPERATIONS. 

«,*  +  *,.? +  *,*...  =  JVi;i 

tf£  +  ££+£  '.'.'.  =  N,\  \  '     '     '     '    (A) 

*!.,:=:  j£J 


in  which  the  coefficients  are  known.  As  the  number  of  un- 
known quantities  is  less  than  the  number  of  equations,  a  direct 
solution  is  impossible. 

Designating  the  errors  by  u,  we  can  write  equations  at  (A), 


-  Nn  =  un. 


By  the  principles  already  stated,  the  most  probable  values 
for  these  various  quantities  are  those  which  render  the  sum  of 
the  squares  of  the  errors,  z/,2  -f  it*  .  .  .  -\-n,?,  a  minimum.  Plac- 
ing all  terms  but  those  depending  upon  x,  equal  to  Mlt  Mt 
.  .  .  MM,  equations  at  (B)  will  take  the  form 

a^x  -f-  MI  =  ul  ;  -j 

atx  -f  M*  =  «,  ;  I  / 


Taking  the  sum  of  the  squares  of  both  members  of  the  equa- 
tions at  (C),  we  obtain 


J...  +  (anx  +  3f.)«  =  «/+  «,a  .  .  .  «.'. 
Differentiating  this  with  respect  to  ^r,  and  placing  the  first  dif- 


CALCULATION  OF   THE    TRIANGULATION.  155 

ferential  coefficient  equal  to  zero,  we  have,  after  dividing  by  2, 
'/    " 

From  this,  we  see  that  to  form  the  most  probable  value  for  x, 
we  multiply  each  equation  by  the  coefficient  of  x  in  that  equa- 
tion, add  these  products,  and  place  the  sum  equal  to  zero.  By 
doing  this  with  y  and  z  we  will  obtain  one  equation  for  each 
unknown  quantity,  from  which  each  can  be  found  by  the  or- 
dinary methods  of  elimination. 

To  illustrate  :        x-\-2y  -f-  2z  —  2  =  o  ; (i) 

—  2x  +  y+   ,?+ 4  =  0; (2) 

3*  +  y-    ,3- -3  =  0; (3) 

x  —  2y  -f-  2z  —  8  =  o.  .     .     .    .    .     .  (4) 

Multiply  (i)  by  I,  x -\-2y-\-2z  —    2=0; 

multiply  (2)  by  —  2,  4*  —  2y  —  2z  —    8=0; 

multiply  (3)  by  3,  gx  -f-  37  —  3-3-  —    9  =  0; 
multiply  (4)  by  i,  x  —  2y  -f-  2z  —    8=0; 

by  adding,  15-^+   y  —    3  —  27  =  o (5) 


Multiply  (i)  by  2,  2;tr  -f-  47  +  4#  —    4  =  o  ; 

multiply  (2)  by  i,  —  2^- -(-  jj/+    <sr  +   4  =  0; 
multiply  (3)  by  i,  3^+J—    ^—    3=o; 

multiply  (4)  by  —  2,  —  2.x  -f-  4JK  —  4^  +  16  =  o  ; 
by  adding,  x-{-ioy  -|- 13  —  o.      ...     (6) 

Likewise  by  multiplying  (i)  by  2,  (2)  by  I,  (3)  by  —  I,  and  (4) 
by  2,  and  adding,  we  get 

—  X+  102-  13  =  0 (/) 


1 56 


GEODETIC  OPERATIONS. 


Eliminating  (5),  (6),  and  (7),  which  are  called  the  normal 
equations,  by  the  usual  algebraic  method,  we  find  x  =  2,  y  = 
—  1.5,  and  z—  1.5. 

To  further  illustrate  this  method,  we 
H.K  will  take  another  example  : 


C.T 


FIG.  16. 


(i) 


=    87°  47'  42".s  ; 


(a)    H.KtoH    =  144  17  47  .5; 

(3)  C      to  77     =     56  30  09  .o  ; 

(4)  £7       to  C.T  =  148  04  22  .5; 

(5)  H     to  C.T  =    91  34  14  -SI 

(6)  C.TtoH.K=i24  07  29.5. 

In  this  figure  there  are  three  condi- 
tions to  be  fulfilled  : 


(3)  +  (5)  =  (4),  (O  +  (3)  =  (2),  and  (2)  +  (5)  +  (6)  =  360°. 

As  the  changes  in  these  values  will  not  likely  affect  any- 
thing beyond  the  seconds,  suppose  we  designate  the  seconds 
of  the  angles  by  a,  b  .  .  .  f,  so  that  we  will  write  the  angles  : 

(i)=    87°  47'  +  *"; 
(2)  =  144   17  +b  ; 

(6)  =  124  07  +/  . 


(3)  +  (5)  =  (4),    (3)-    56° 

(5)=    9i°  34'+'; 
(3)  +  (5)  =  H80  04'+  c  +  e,      (4)  =  148°  04'  +  d  ; 


therefore, 


CALCULATION  OF  THE    TRIANGULATION.  157 

Also  (I)  +  (3)  =  (2),  (i)  =  87°47'+  a  ; 
(3)=  56°30'+^ 
(i)  +  (3)  =  I44°i7'+  *  +  c,  (2)  =  144°  i/  +  b  ; 

therefore,  a  -j-  £  =  b. 

(2)  +  (5)  +  (6)  =  360°,  (2)  =  144°  17'+  b  ; 

(5)=    9i   34  +  *; 
(6)=  124  07+/; 
(2)  +  (5)  +  (6)  -359°  58'  +  £  +  *+/=  360°; 

therefore,  £+  ^  +/=  120". 

By  observation,  #  =  42".5  ; 

b  =  47"-5  J 

'  =  09"   ; 

^  =  22".5  ; 

*  =  i4".s  ; 

/=  29".5. 
From  condition,  c-{-e  =  d\ 


Substituting  in  observation  equations  the  values  of  d,  and  b 
as  determined  by  the  conditional  equations,  we  can  write 

a=  42".$; 

a  +  c=  47".$  (a  +  c)  =  6: 

c=  9"  ; 

c  +  e=  22".$  (c  +  e)  =  d; 

e=  i4".5; 

/=  29".5; 

=  I207'. 


158  GEODETIC  OPERATIONS. 

From  these  we  obtain  the  following  normal  equations : 

2a  +  c  =  90 ; 
=  120; 
=  79; 

=i57; 
fi  +  e  +  2f  =  149.5. 

Solving  for  a,  b,  ct  e,  and/",  by  ordinary  method  of  elimina- 
tion, we  find  a=  4i"-i25,  b  =  48".8;5,  £  =  7".75,  ^=<r  +  ^ 
=  49"-375>  '  =  4i"-625,  and/  =  29".5. 

This  gives  for  the  angles  the  following  as  the  most  probable 
values : 

(i)=    87°  47'  41 ".  1 25; 

(2)  =  144  17  48  .875  ; 

(3)  =  56  30    7  75  ; 

(4)  =  148  04  49  .375  ; 

(5)  =  91  34  4i  -625  ; 

(6)  =  124  07  29  .5. 

Observations  with  different  weights  can  be  adjusted  by  this 
method.  Since  we  do  not  use  in  this  case  the  square  of  the 
error,  or  some  quantity  involving  the  error  squared,  but  only 
the  first  power,  we  must  therefore  multiply  the  error,  or 
quantity  involving  the  error,  by  the  square  root  of  the  weight. 
The  weight  can  be  determined  from  the  probable  error  as  ex- 
plained on  page  123,  if  not  taken  directly  from  the  number  of 
measurements. 

When  it  is  desired  to  make  use  of  this  method  for  adjusting 
observations  of  different  weights,  the  outline  of  the  method 
may  be  given  as  follows. 

For  each  of  the  observations  write  an  observation  equation. 


CALCULATION  OF  THE    TRIANGULATION.  159 

For  each  condition  write  a  conditional  equation. 

From  the  conditional  equations  obtain  as  many  values  as 
possible  for  one  unknown  quantity  in  terms  of  others,  and  sub- 
stitute in  the  observation  equations.  Multiply  each  observa- 
tion equation  by  the  square  root  of  its  weight.  Form  the  nor- 
mal equations  and  solve  as  in  ordinary  cases. 

While  normal  equations  will  afford  an  excellent  solution  for 
any  number  of  observation  and  conditional  equations,  the 
labor  becomes  quite  great  when  we  have  a  large  number  of 
equations,  or  large  quantities  to  handle. 

In  such  cases  the  method  of  correlatives  as  developed  by 
Gauss  will  afford  the  readiest  solution.  This  method  pertains 
to  equations  of  condition  only,  and  in  terms  of  corrections  that 
are  to  be  applied  to  the  various  quantities  in  order  to  make 
them  fulfil  the  required  conditions. 

Suppose  a,  ft,  y  .  .  .  represent  the  corrections,  and  the  con- 
ditional equations  expressed  in  terms  of  these  corrections  with 
coefficients  whose  values  are  known,  as  well  as  the  absolute 
term  ;  for  instance,  in  the  last  example  we  had  the  condition 
(3)  +  (5)  -  (4),  but  in  reality  (3)  +  (5)  =  (4)  +  i",  or  (3)  +  (5) 

—  (4)  =  i".     So  if  a-,  /?,  and  y  represent  the  corrections  ap- 
plied to  (3),  (4),  and  (5),  their   algebraic   sum  should  equal 

—  i",  to  counteract  the  error  -{-  i";  that  is,  a  -j-  y  —  /3  =  —  i". 
In  this  case  the  known   coefficients  are  i,  and   the  absolute 
term  —  \"  .     So  that,  in  general,  we  may  express  the  condi- 
tional equations  in  terms  of  known  coefficients,  and  absolute 
terms,  with  the  corrections  as  the  unknown  quantities  ;  as 

ava  +  a,fi  -f  a,y  .  .  .  =  M,  ; 


,a  -f  n$  -f  n3y  .  .  .  =  Mn. 


I6O  GEODETIC  OPERATIONS. 

Since  the  most  favorable  results  are  obtained  by  making  the 
sum  of  the  squares  of  the  errors  a  minimum,  if  we  take  a^ 
+  a*0*  +  a*Y*  •  •  •  an9^  =  M,  and  differentiate  it  with  respect 
to  each  variable,  and  making  the  first  differential  equal  to  zero, 
we  will  have,  after  dividing  by  2, 

jlfi  -{-  a^dy  .  .  .  +  andg>  =  o  ; 

jy  ...  + 


m^dft  +  m^dy  .  .  .  -f-  mnd<p  —  o. 


Also,  «a  +  f?  +  Y*  •  •  •  +  <P*  =  a  minimum,  or 

a  .da.-^-  ft  .  d/3  -j-  y  .  dy  .  •  •  +  q>  •  dg>  =  o.    .     .     (B) 

As  the  number  of  equations  is  less  than  the  number  of  un- 
known quantities,  a  part,  as  M,  can  be  found  in  terms  of  the 
others  ;  with  these  values  substituted  in  equations  (A),  we  will 
have  M  less  than  originally,  and  each  of  these  may  be  made 
equal  to  zero.  Chauvenet  accomplishes  this  result  in  the  fol- 
lowing way:  multiply  the  first  equation  at  (A)  by  £,,  the 
second  by  kv  the  third  by  &3,  and  the  nth  by  kn  and  equation  (B) 
by  —  I  ;  then  add  these  products.  Now,  supposing  £,,  k^  .  .  . 
etc.,  are  such  that  M  of  the  differentials  disappear,  the  final 
equation  will  contain  M'  —  M  (calling  M'  the  original  number) 
differentials  with  M'  equations.  Making  them  severally  equal 
to  zero,  we  get 

c,k3  .  .  .  m,km  —  a  =  o  ; 


ankn  -f  bnkn  +  cnkn  .  .  .  mnkn  -  cp  =  o. 
Now,  by  multiplying  the  first  by  av  the  second  by  a^  etc.,  and 


CALCULATION  OF   THE    TRIANGULATION. 


l6l 


adding  the  products,  expressing  the  sum  of  like  terms  by  2, 
we  get 


Likewise,  multiplying  the  first   equation  by  blt  the  next  by 
^  .  .  .  bny  we  have 


,  -f 


This  will  give  as  many  normal  equations  as  there  are  unknown 
quantities  £„  kv  etc.;  so  that  we  obtain  a,  ft,  y,  etc.,  in  terms 
of  £„  kv  etc.  While  the  theory  of  this  is  quite  complicated 
and  involves  a  knowledge  of  differential  equations,  in  practice 
it  is  exceedingly  simple,  as  the  appended  example  will  show: 


44; 
.01  ; 
.46; 


(1)  B.RioC=    75°  3i'  53' 

(2)  B.R  to  R  =  144  36  49 


(3)  B.R  to  £  =  239  35  03 

(4)  C     to  R  —    69  05  oo  .57  ; 

(5)  C      toG  =  164  02  51  .52; 

(6)  R     to  G  —    94  58  05  .44. 

The  conditions  to  be  fulfilled  are : 

(i)  +  (4)-(2)=o; 
-(3)=o; 


B.R 


FIG. 


However,  we  find  that 


(I) +  (4) -(2)=         5" 
(2) +  (6) -(3)=-    9".oi 
(4) +  (6) -(5)=       i4"-49 


162 


GEODETIC  OPERATIONS. 


and  the  corrections  necessary  to  neutralize  these  errors  will  be 
—  5,  -J-Q.OI,  and  —  14.49-  Indicating  the  corrections  by  the 
same  symbols  we  have  used  for  the  angles,  and  transposing  the 
constant  needed,  we  will  write  the  above  equations, 

(i) +  (4) -(2)+  5  =o,  (a) 
(2) +  (6) -(3)-  9.01-0,  (b) 
(4) +  (6) -(5) +14.49=0,  (c) 

Now  we  rule  as  many  vertical  columns  as  there  are  con- 
ditions in  this  case,  three,  and  as  many  horizontal  ones  as  there 
are  quantities  to  correct, — in  this  case  six. 


In  the  first  condition  we  have  -j-  (i) 
-f-  (4)  —  (2),  so  we  write  -|-  k^  opposite 
I,  -\-  kl  opposite  4,  and  — kl  opposite 

2. 


The  second  condition  has  (2)  +  (6)  —  (3);  so  we  put  -j-  &, 
opposite  2,  and  6,  and  —  &,  opposite  3. 

The  third  condition  involves  (4)  -\-  (6)  —  (5);  so  we  put  -f-  k* 
opposite  4  and  6,  and  —  £s  opposite  5. 

The  first  equation  of  correlative  is  to  contain  the  contents  of 
the  1st  and  4th  horizontal  columns,  and  minus  the  contents  of 
the  2d  ;  this  is  determined  by  that  equation  having  (i)  -(-  (4) 


1st   column  contains    -J-^,; 

2d    column  contains    -j-  ^»  ~~  &v  [signs  changed  as  it  is  —  (2)] 

4th  column  contains    -f~  £,  -f-  &,; 

ist  correlative  contains  3^  —  £,  -f-  £s. 


CALCULATION  OF   THE    TRIANGULATION. 


163 


In  the  second  conditional  equation,  we  have  (2)  -j-  (6)  —  (3), 
so  we  take  the  contents  of  the  2d  and  6th  horizontal  columns 
and  the  3d  with  signs  changed. 

2d  contains  —  kr  -f-   £,          ; 

6th  contains  -(-   k^-\-k^ 

3d  contains  (sign  changed)  -(-   &,          ; 

(2)  +  (6)  -  (3)  contains  -  k,  +  &  +  kv 

2d  correlative  equation. 


Likewise  for  the  3d  we  get  £» 

Placing  these  correlatives  in  the  equations  involving  the  cor- 
rections, (a),  (t>),  and  (c),  we  get 

3^,  -£,+   £,+    5       =0; 

—  £,  +  3/&,  +   £,  —    9-01  =  o  , 

£.+   4  +  3^  +  14-49  =  o. 

By  ordinary  process  of  elimination,  we  find  £,  =  3//-37,  k^  = 


Angle. 

ISt. 

2d. 

3d. 

Correction. 

Corrected  Angles. 

I 

+.-A 

+  337 

75°  31'  56".8i 

2 

4^*i 

+  ^a 

-  3.37  +  687 

144   36  52  .51 

3 

-    ^2 

-  6.87 

239   34  56  .58 

4 

-fi, 

+  /&S 

+  3-37  -  8.24 

69  04  55  .70 

5 

->&3 

+  8.24 

164  02  59  .76 

6 

**• 

+  4i 

+  6.87-8.24 

94   58  04  .07 

These  corrections  are  determined  in  this  way : 

(1)  is  in  the  first  condition  and  positive,  so  it  is  affected  by 

-{-*.; 

(2)  is  in  the  first  and  second, — negative  in  the  first,  and  posi- 

tive in  the  second ;  therefore  it  is  affected  by  —  k,  and 


164  GEODETIC  OPERATIONS. 

(3)  is  negative  in  the  second  so  it  is  corrected  by  —  kt  ; 

(4)  is  positive   in  the  first  and  third,  so  its  correction  will  be 

^M»-MM 

(5)  is  negative  in  the  third,  therefore  its  correction  is  —  kt  ; 

(6)  is  positive  in  the  second  and  third,  so  it  will  be  corrected 

by  +*,  +  *,. 

These  values  of  £,,  kv  and  k^  applied  as  just  indicated  to  the 
observed  angles,  will  give  the  most  probable  values  for  the 
angles  that  will  make  them  conformable  to  the  conditions. 

It  may  be  noticed  that  the  method  of  forming  the  equa- 
tions of  correlatives  is  the  same  as  forming  normal  equations. 
To  illustrate,  let  us  take  (a)  of  the  conditional  equations  ;  the 
coefficient  of  (i)  is  -{-  I,  of  (4)  is  -}-  I,  and  of  (2)  is  —  i. 


Multiply  horizontal  column  i  by  -f  i,  =  -j-  A 
multiply  horizontal  column  4  by  -f-  I,  =  -}-  k, 
multiply  horizontal  column  2  by  —  i,  =  -\-  kl 

(I)  +  (4)  -(2)  =       3^ 

therefore  3^,  —  &,  -f-  &,  +  5  =  O. 


This  is  the  better  plan  when  the  coefficients  are  not  unities, 
When  the  observations  have  different  weights,  the  operation 
is  somewhat  complicated  and  can 
be  best  explained  by  solving  an 
example  ; 

(1)  Cto  P  =  107°  53'  oo".  07  weight  5; 

(2)  Cto  A  =  171  42  02  .18  weight  4; 

(3)  Cto  B  =  198   10  28  .22  weight  6; 
(4)Pto  A  =    63  49  05  .86  weight  2; 
($)Pto£  =    90  17  16  .02  weight  3; 
(6)  A  to  B  =    26  28  04  .54  weight  i. 


CALCULATION  OF   THE    TRIANGULATION. 
The  conditional  equations  are  : 


i65 


)  +  (5)  -(3)  -12.13-0; 
-(5)-    5-62=0; 


designating  the  corrections  by  the  same  symbols  as  the  angles. 
If  the  equations  on  page  159  had  been  weighted  before  dif- 
ferentiation, a*,  (?...(}?  would  have  been  multiplied  by  the 
respective  weights  of  the  observation  to  which  they  were  to 
form  corrections.  These  weights,  say  wlt  «>„...  «/„,  being  con- 
stant factors,  would  remain  in  the  differentials ;  so  that  the 
equations  just  referred  to  would  have  for  their  last  term  —  w,or, 
—  iv^ft  ...  —  wn<t>.  Then  afterwards,  when  multiplied  by  #„  av 
etc.,  before  summing  the  products,  in  order  to  get  or,  ft  .  .  .  <p 
freed  from  factors,  since  we  only  know  the  values  of  these 
errors  unaffected  by  their  weights,  we  must  divide  the  first 
equation  by  wlt  second  by  wv  etc. 


We  make  the  arrangement  as  though 
there  were  no  weights  so  far  as  the 
position  and  signs  of  the  correlatives 
are  concerned,  but  take  the  reciprocal 
of  the  weight  of  the  angle  as  the  co- 
efficient. 


Angle. 

*,. 

*,. 

*,. 

I 

+  t*i 

+  i^3 

2 

—  i>£j 

3 

—  ^2 

4 

+  i*. 

+  i^s 

5 

~f~  "i"^2 

—  i^3 

6 

+  1*3 

To  illustrate :  the  first  condition  equation  involves  a  correc- 
tion to  be  applied  positively  to  (i)  and  (4),  and  negatively  to 
(2).  And  since  the  reliability  of  these  angles  is  proportional 
to  5,  2,  and  4,  it  is  apparent  that  the  corrections  they  should 
receive  would  be  in  the  inverse  proportion,  or  ^,  £,  and  £. 


1  66  GEODETIC  OPERATIONS. 

Therefore  for  the  correction,  this  equation  suggests  that  for 
(l)  we  should  write  ££,;  for  (4),  £/£,;  and  for  —  (2),  —  \k^. 

Second  conditional  equation  involves  corrections  to(i)-|- 
(5)  —  (3).  These  angles  are  in  point  of  accuracy  proportional  to 
their  weights,  5,  3,  and  6;  therefore  the  corrections  will  have 
the  inverse  proportion,  \,  ^-,  and  \.  So  we  write  the  correc- 
tions ;  the  second  conditional  equation  suggests  for(i),  -f-  \k^ 
for  (5),  +R:  and  for  -  (3),  -  \kv 

Likewise  in  the  third,  for  (4),  -\-\k^  for   (6),  -f-  \kt\  and  for 


Now,  to  form  the  equations,  the  first  condition  requires  the 
sum  of  the  quantities  in  the  first  and  fourth  horizontal  col- 
umn, and  the  negative  of  the  second. 


(i)  contains  j 

(4)  contains  £ 

—  (2)  contains  \k^ 

(i)  +  (4)  -  (2)  contains  (|  +  i  +  *)*,  +  \k%  + 


The  second  condition  requires  (i)  +  (5)  —  (3). 

(i)  contains  ^kl  -\-  \kt 

(5)  contains  -|~  %k%  —  f^3; 

—  (3)  contains  -J-  %kt  ; 

(0+  (5)  ~  (3)  contains  ^  +  (|  +  ^  +  ^s  -  ^,.      (/) 


Likewise,  (4)  contains  ££, 

(6)  contains  +    &,; 

—  (5)  contains        —  ££,  -f  \k» 

(4)  +  (6)  -  (5)  contains  -R  -  \k%  +  (i  +  i  +  t)^,-     (^) 

Clearing  equations  (e\  (/),  and  (g)  of  fractions  and  substi- 


CALCULATION  OF  THE    TRIANGULATION. 


167 


luting  them  for  the  values  of  the  corrections  in  (#),  (£),  and  (c), 
we  get 

375=o;  ....  (A) 
--^-  12.13  =  0;  .  .  . 

5-62-0;  ... 


(0 


Reducing  57^,  -f-  I2&,  +  3o£s  +  225.00  =  o  ; 
6k,  -j-  2i£,  -  io£s  -  363-9°  =  °  5 
3^,  -  2&1+ii£,  —  33.72  =  0. 

Eliminating  by  the  usual  process,  we  find  that 

k,  —  —  i6".so,        4  =  28".o8,        £,  =  I2".6S. 
The  plan  for  applying  these  values  can  be  best  exhibited  : 


Angle. 

ISt. 

zd. 

3d- 

Correction. 

Corrected  Angle. 

2 

-tt 

i^3 

-  3-30+  5-6l 
-  (-4-12) 

107°  53'  02".  38 
171   42  06  .30 

3 

-M, 

-  4.68 

198   10  23  .54 

4 

i*. 

|>&3 

-  8.25+6.34 

63   49  03  .95 

5 

Mi 

-**• 

9.36  —  4.22 

90     17    21    .16 

6 

*, 

12.68 

26    28    17    .22 

(1)  is  corrected  by  ££,  and  \kv  or  —  3.30  +  5-6i  ; 

(2)  is  corrected  by  —  •££,  or  —  \(—  16.50)  =4.12,  etc. 

In  the  case  of  weighted  observation,  the  method  of  correla- 
tives is  far  the  simplest. 

While  station  adjustment  is  of  somewhat  frequent  occur- 
rence, yet  the  angles  regarded  as  forming  parts  of  a  triangle 
more  frequently  require  attention.  The  geometric  require- 
ment that  the  three  angles  of  a  triangle  equal  180°  furnishes  a 
condition  to  begin  with  ;  likewise,  these  angles  as  individuals 


1 68  GEODETIC  OPERATIONS. 

may  form  a  part  of  a  station  condition.  In  this  case — which  is 
the  rule — we  combine  station-adjustment  with  what  is  known 
as  figure-adjustment;  that  is,  bringing  the  angles  into  con- 
formity with  the  geometric  requirements  of  the  figure. 

The  first  geometric  condition  is  that  the  angles  of  the  tri- 
angles equal  i8o°— |-  spherical  excess,  or  A -\-  B -\- C  —  £  = 
180°,  in  which,  A,  B,  and  £7  are  the  measured  angles  of  the 
triangle,  and  e  =  spherical  excess.  To  find  what  the  errors 
are  in  the  case  of  each  triangle,  it  is  necessary  to  determine 
the  value  of  £.  By  geometry  we  know  that  the  three  angles  of 
a  spherical  triangle  bear  the  same  relation  to  four  right  angles 
that  its  area  bears  to  a  hemisphere  ;  that  is,  £  :  27r::area  :  27rra, 

area  area 

f  =  — r-.    £  being  small,  £  in  seconds  =  £  .  sin  I  .  £  =  -^ — : .-.. 

r  r3  sin   i" 

As  the  triangle  is  small  compared  with  the  surface  of  the 
sphere,  it  may  be  regarded  as  equivalent  to  a  plane  triangle  = 

^a.b .  sin  C,  hence  £  =  — i — : — •— ;,  in  which  a,  b,  and  C represent 

the  two  sides  and  included  angle,  and  r  the  radius  of  a  sphere. 
r  can  be  considered  a  mean  proportional  between  the  radius 
of  curvature  of  a  meridian  and  the  normal  of  a  point  whose 
position  is  the  centre  of  the  triangle. 

On  page  207,  R  —  -, ,   .  ,  ,XJ>  N  =  i —  •  »  rci »  r*  = 

(i  —  f  sin*  L}*  (I  —  **  sin*  £)* 

N .  R  =  -, — _    a  ~  a  y-\i  I  dividing  by  I  —  ?  and  neglecting  terms 

^i 
involving  powers  of  e  above  the  fourth,  r"  = 


_  2f  sin1  L 


d  & 

.  .,  J-N  =  — ; — a F-     Substituting  this  val- 

—  2  sm'Z)       I  +  e  cos  2L 


f          a.b&m 

ue  for  r\  £  =  —         — j-r — T, -.  The  factor . 

2a  sm  I  2«asin  i" 

varies  with  2Z,,  and  can  be  computed  with  L  as  the  variable 
for  every  30'  and  tabulated  ;  calling  this  term  «,  £=a .  b .  sin  C .  n. 


CALCULATION  OF   THE    TRIANGULATION. 


[69 


The  most  elaborate  spheroidal  triangle-computation  for 
spherical  excess  shows  that  the  result  obtained  by  using  the 
above  formula  will  differ  from  the  correct  value,  only  in  the 
thousandth  part  of  a  second.  For  preliminary  field  computa- 
tion the  excess  may  be  taken  as  i"  for  every  200  square  kilo- 
metres, or  75.5  square  miles  ;  and  when  the  sides  are  4  miles  or 
under,  it  can  be  disregarded.  The  following  table  contains  n 
for  L,  from  24°  to  53°  30',  based  upon  Clarke's  spheroid.  The 
table  must  be  entered  with  the  average  latitude  of  the  tri- 
angle approximately. 


Latitude. 

Log*. 

Latitude. 

Log*. 

Latitude. 

Log*. 

24°  oo' 

I.40596 

34°  oo' 

1.40509 

44°  oo' 

1.40410 

24  30 

92 

34  30 

05 

44  30 

05 

25  oo 

83 

35  oo 

00 

45  oo 

00 

25  30 

84 

35  30 

495 

45  30 

395 

26  00 

80 

36  oo 

9i 

46  oo 

90 

26  30 

76 

36  30 

86 

46  30 

85 

27  oo 

72 

37  oo 

81 

47  oo 

80 

27  30 

68 

37  30 

76 

47  30 

75 

28  oo 

64 

38  oo 

7i 

48  oo 

69 

28  30 

1.40559 

38  30 

I  .  40466 

48  30 

1.40364 

29  oo 

55 

39  oo 

61 

49  oo 

59 

29  30 

5i 

39  30 

56 

49  30 

54 

30  oo 

47 

40  oo 

5i 

50  oo 

49 

30  30 

42 

40  30 

46 

50  30 

44 

31  oo 

37 

41  oo 

4i 

51  oo 

39 

31  30 

33 

41  30 

36 

51  30 

34 

32  oo 

28 

42  oo 

3i 

52  oo 

29 

32  30 

24 

42  30 

26 

52  30 

24 

33  oo 

19 

43  oo 

20 

53  oo 

19 

33  30 

1.40514 

43  30 

I.404I5 

53  30 

1.40314 

The  spherical  excess  computed  by  this  formula  is  for  the 
entire  triangle ;  and,  unless  there  is  considerable  difference  in 
the  lengths  of  the  opposite  sides,  one  third  of  the  excess  is  to 
be  deducted  from  each  angle  of  the  triangle ;  but  this  reduced 
value  is  used  only  in  the  triangle  condition,  and  not  in  the 
station  condition. 


170 


GEODETIC  OPERATIONS. 


If  in  this  figure  we  have  measured  the  angles  numbered  and 
have  the  averages,  which  we  will  designate  by  the  numbers,  it 
will  be  seen  that  a  great  variety 
of  conditions  may  be  written. 
But  upon  examination  it  will  be- 
come apparent  that  some  of  th'e 
angles  are  indirectly  two  or  more 
times  subjected  to  the  same  or 
equivalent  conditions.  For  in- 
stance,  if  (3)  +  (7)  +  (9)  =  180°, 
and  (ii)  +  (13)  +  (i)  =  1 80°,  the 
condition  that  (2)  +  (7)  -f-  (10)  -f- 
(13)  =  360°  is  already  fulfilled. 
Also,  if  (i) +  (3) -(2),  and  (2)  + 
(5)  =  (4),  the  condition  (i)  +(3)  +  ($)  =  (4)  is  unnecessary; 
and,  again,  if  (3)  +  (7)  +  (9)  =  i*)0,  (i  i)  +  (13)  +  (0  =  180°, 
and  (14)  +  (2)  -f-  (6)  =  180°,  then  (i)  +  (3)  =  (2)  is  unneces- 
sary. If  we  have  the  most  probable  value  for  (6)  and  (7),  their 
difference  will  be  (8)  without  involving  (8)  in  the  adjustment; 
or  if  we  have  the  best  values  for  (8)  and  (6),  their  sum  will 
give  (7) ;  or  if  we  have  (i),  (2),  and  (4),  we  can  find  by  subtrac- 
tion the  most  probable  value  for  (3)  and  (5). 

From  this  we  learn  that  it  is  useless  to  involve  whole  angles 
and  all  of  their  parts  in  different  conditions.  With  such  a  fig- 
ure the  following  conditions  would  be  sufficient : 


FIG.  19. 


(2) +  (5) -(4); 

(3) +(7) +  (9) -i  80°; 

(8)  +  (10)  +  (12)  =180°; 

(2)  + (7)  + (10) +  (13) -360°; 

(13) 


Other  combinations  could  also  be  used. 


CALCULATION  OF   THE    TRIANGULATION. 


In  such  adjustments  the  method  of  correlatives  should  be 
used,  as  the  labor  does  not  increase  rapidly  in  proportion  to 
the  increased  number  of  conditions. 

The  equations  like  those  given  so  far  are  called  angle  or  an- 
giilar  equations.  The  theorem  in  trigonometry  that  the  ratio 
of  sides  is  equal  to  the  ratio  of  the  sines  of  the  opposite  angles 


gves  us  -j-~  = 


.     .     . 


Since  A,  B,  and  C  are  fixed  points, 


the  distances  AB  and  AC  are  constant;  therefore  -^repre- 
sents a  constant  quantity,  so  that  if  (3)  and  (7)  are  changed  at 
all,  the  sine  of  (3)  and  its  correction 
must  have  the  same   ratio  to  the 
sine  of  (7)  and  its  correction  that  sin 
(3)  has  to  sin  (7).      This  involves 
another  condition,  which  will  now 
be  elaborated. 

From  the  theorem  just  referred 
to,  we  obtain  the  following  equa- 
tions : 


OB  _  sin  bt         (M  _  sin  b,         OD  _  sin  bt        OC  _  sin  bl 
OA        sin  a/        OD  ~~  sin  a,'        OC  ~  sin  «,'       OB  ~  sin  «/ 


Multiplying  these  equations  together,  member  by  member,  we 
obtain 


OB .  OA  .  OD  .  OC  _     _  sin  bt .  sin  b, .  sin  bt .  sin  b, 
OA  .  OD  .  OC.  OB  ~      ~  sin  a, .  sinaa .  sin  a, .  sin  aA 


or,     sin  #, .  sin  a, .  sin  a3 .  sin  at  =  sin  bt .  sin  ba .  sin  b3 .  sin  b^ 


1/2  GEODETIC  OPERATIONS. 

But  these  are  the  values  after  correction,  so  we  will  put  M^ 
Mv  Mv  Mt,  for  alt  a,,  av  a;,  Nlt  Nv  Ntt  and  N.  for  £„  bn,  bv 
and  b;,  and  denote  the  necessary  corrections  by  vlt  v,,  v^  v4,  and 
x^xv  x3,  and  x4.  Substituting  these  values  in  the  last  equa- 
tion, we  have 

sin  (M,  +  f ,) .  sin  (M3  -f  v,) .  sin  (M,  +  z\) .  sin  (J/4  +  z/4) 
=  sin  (N,  +  ^) .  sin  (Nn  +  ^) .  sin  (TV,  -f  ^r8) .  sin  (N4 


or,  passing  to  logs, 

log  sin  (M,  +  ?,)  +  log  sin  (^f,  -f-  v,)  +  log  sin  (M,  +  »,) 

+  log  sin  (Mt  -f-  ^4) 
=  log  sin  (^  +  *,)  +  log  sin  (^2  +  xt)  +  log  sin  (N3  +  ^r3) 


Since  vv  vv  v3,  v<,  x»  xv  x^  x^  are  very  small,  we  may  develop 
each  of  the  above  terms  by  Taylor's  theorem,  stopping  with 
the  first  power  of  the  correction  : 


,    (d  log  sin  M.\ 
log  sin  (M,  +  Vl)  =  log  sin  M,  +   -  ^1**  '> 


,.,,...  ,  sn      . 

log  sin  (M,  +  v.)  =  log  sin  M%  +    -  *jg* 

etc.,  etc.; 

Id  log  sin 


.    ,,,  I 

log  sin  (N,  +  ^)  =  log  sin  N,  +  ( 


etc.,  etc.; 

in  which  ?>,,  vv  vt,  z/4,  ^r,,  ^rt,  x9,  x»  are  expressed  in  seconds,  so 


CALCULATION  OF   THE    TRIANGULATION.  173 

that  --  -rrf  -  1  is  the  log  difference  for  one  tabular  unit  for 

the  angle  Mv  or  the  tabular  difference  for  Mt;  let  us  call  this 
difference  d»  dv  dt,  d»  for  M»  Mv  Mv  Mtt  and  #„  tfa,  tfs,  #4,  for 

A7      AT1      AT     ]\f 
*»jj  -<va>  ^V3>  •"•« 

Substituting  these  values  in  the  lr.st  equation,  we  have 


log  sin  Ml  -f-  d?X  -f-  log  sin  Mt  -\~  dju^  -|-  log  sin  J/3 

-f-  log  sin  J/4 
=  log  sin  N,  +  tf,^,  +  log  sin  7V,  -f  d,^,  -f-  log  sin  7V3  +  6^t 

-j-  log  sin  ^V4-[-  ^4^r4. 

When  transposed, 

log  sin  Ml  +  log  sin  J/2  -(-  log  sin  J/3  -f-  log  sin  M< 
—  log  sin  A7,  —  log  sin  N^  —  log  sin  N3  —  log  sin  Nt 

=  to  +  to  +  to  +  to  -  <ta  -  «te  -  ^3^3  -  <»4 

—  an  equation  in  which  the  unknown  quantities  are  the  correc- 
tions to  the  angles,  or  the  same  quantities  that  are  sought  in 
the  adjusting  equations. 

This  gives  directly  an  equation  of  condition,  for  since  the 
sum  of  the  log  sines  of  Mt,  M^  M3,  and  Mt  should  equal  the 
sum  of  the  log  sines  of  N^  N^  N^,  and  Nv  the  corrections  d1vl 
+  df>i  +  dtvt  +  dpt  should  equal  dlX,  -\-  8^  -f  d^  +  djcc 
But  if  the  log  sines  of  (M)  differ  from  the  log  sines  of  (TV),  then 
that  amount  of  difference  must  be  corrected  in  ef]vl  -j-  djtt  .  .  . 
etc.  This  is  called  the  linear  equation. 

By  way  of  illustration,  suppose  we  have  the  appended  fig- 
ure with  the  average  angles  as  given  : 

(i)=  5o°3i'i3".68; 
(2}=  14  51  47  .88; 
(3)  not  needed  ; 


174 


GEODETIC  OPERATIONS. 


(4)= 
(5)= 
(6)  = 
(7) 
(8)  = 


71°  46'  16-36; 
82   32  49  .52; 
32   04  12  .49; 
not  needed  ; 
30   03   29  .39; 


(9)  =133   03  52  .48; 
(10)  =    67   23  18  .99; 
(n)          not  needed; 
(12)  =     57   42  49  .56. 

We  first  deduce  the  linear  equa- 
tion : 


//#.sin(io)  =  HP.  sin  (8); 


by  multiplication1,. 

sin  (2) .  sifl  (lo) .  sin  (4)  :=  sin  (6)  .  sin  (g)  «  sin  (12)* 

Writing  for  tabular  difference  <?(2),  <?(lo),  etc,  and  [2],  [10], 
etc.  as  the  corrections  for  (2),  (10),  etc.,  we  have 

log  sin  (2)  +  8(2}  [2]  +  log  sin  (lO)  -f  «J(iO)  [10] 

-{-logsin(4)  +  d(4)[4] 
=  log  sin  (6)  +  <J(6)  [6]  +  log  sin  (8)  -f  «T(8)  [8] 

+  log  sin  (12)  -f  <?(i2)  [12], 

From  the  table  of  logs,  we  find  : 

log  sin    (6)  =    9.72505722,         8  (6)  =  .00000336; 
log  sin    (8)  =    9.69978200,         S  (g)  =  364 ; 

log  sin  (13)  =    9.92705722,         6(l2]=  133; 

sum  =  29.35189644. 


CALCULATION  OF   THE  TRIANGULATION.  175 

log  sin     (2)  =     9.40910559,  d   (2)  =  .00000794; 

log  sin    (4)=    9.97763813,         *  (4)  =  69; 

logsin(io)  =    9-96526395,         tf(io)  =  87; 

sum  —  29.35200767. 


log  sin  (2)  -J-  log  sin  (4)  -(-  log  sin  (10) 

=  log  sin  (6)  -f  log  sin  (8)  -f-  log  sin  (i  2)  -(-  0.00011123. 

As  the  corrections  are  to  neutralize  this  difference,  we  write 


2]  -  O.OOOIII23. 


Substituting  for  #(2),  #(4),  etc.,   their  values,  we   have,  after 
multiplying  by  loooooo  to  avoid  decimals, 


7.94[2]  +  .87[io] 

=  I-33M  4-  3.64[8]  +  3-36[6]  -  111.23. 

Transposing  and  passing  to  our  usual  notation, 

7.94(2)  +  .87(10)  +  .69(4)  -  1-33(12)  -  3-64(8)  -  3-36(6) 

-f-  111.23  =  o. 

The  angle  equations  are  those  involving  the  angles  that 
will  not  be  doubly  adjusted.  In  the  present  case  they  will  be, 
when  expressed  in  terms  of  their  corrections, 

(5)  +  (i)-  (9)  +  10.72-0; 
(9)  +  (2)+  (6)-  7-15  =  0; 
(i)  +  (4)  +  (12)  +19.60  =  0; 
(5)  +  (8)  +  (10)  -22.10  =  0; 


176  GEODETIC  OPERATIONS. 

7.94(2)  +  .87(10)  +  .69(4)  -   1.33(12)  -  3-64(8)  -  3.36(6) 

+  111.23  =  o. 

In  this  (3)  is  omitted,  since  if  (2)  and  (12)  are  known,  (3)  can 
be  found  by  subtraction.  Likewise,  (11)  is  the  sum  of  (8)  and 
(4);  also  (7),  the  difference  between  (10)  and  (6).  So  we  now 
simply  form  the  correlative  equations  from  these  five  condi- 
tional equations. 


ISt. 

zd. 

3d. 

4th. 

5th. 

I 

*i 

*3 

2 

*2 

7-94*6 

4 

k$ 

.  6o>&5 

5 

*i 

*4 

6 

*a 

—  3  36^5 

8 

£4 

-  3^64*5 

9 

—  *i 

*a 

10 

*4 

.87*5 

12 

k3 

-  1.33*6 

The  formation  of  the  first  four  normal  equations  follows  the 
principles  repeatedly  given,  but  as  something  new  may  appear 
in  obtaining  the  fifth  equation,  it  will  be  formed  in  detail. 


7.94  times  column    2  = 

.87  times  column  10  = 

.69  times  column    4  = 

1.33  times  column  12  = 

3.64  times  column    8  = 

3.36  times  column    6  = 

Total, 


7-94^,  +63.04364; 

.87*4+    -7569^; 

.69^  +     .4761^; 

-i-33^  +  1.7689^.; 

-3-64^+13.24964; 
•3-364  +11.28964; 

4.584-  .644-2.774+90.58474. 


Barlow's  table  of  squares  will  facilitate  work,  as  the  coeffi- 
cients of  the  terms  in  the  side  equation  are  squared  in  finding 


CALCULATION  OF   THE    TRIANGULATION. 


177 


the  coefficient  of  the  correlative  corresponding  to  the  equation 
of  condition  formed  by  the  side  equation.  In  this  case,  the 
fifth  conditional  equation  is  the  side  equation,  and  the  coeffi- 
cients of  k6  in  the  fifth  normal  equation  are  the  squares  of 
7.94,  etc. 

The  normal  equations  are  : 


3/fc,  kt  +       £,  +         k<  +    10.72  =  o; 

-    h  +      3^3  +    4-58£6-      7-15  =o; 

+    k,  +     3£8  -      .64^  +    19-60  =  0; 

k,  +       3/&4-    2.77^-      9.29  =  o; 

4.58^  -  .64^,  -  2.77k,  -f  90.58^  +  111.23  =  o. 

The  solution  of  these  equations  gives  kl  =  —  "-53,  &,  =  4//-4O, 
£,  =  —  6".  66,  ki  =  i  ".94,  kb  =  —  i  ".43. 

These  values  are  applied  to  the  various  angles  as  indicated 
in  the  table  just  given.  For  instance,  (2)  is  to  be  corrected  by 
£,  and  7.94  times  kb. 

The  best  rule  that  can  be  given  for  the  formation  of  side 
equations  is  to  regard  one  of  the  ver- 
tices as  the  vertex  of  a  pyramid,  with 
the  figure  formed  by  the  other  points 
as  the  base,  and  take  the  product  of 
the  sines  of  the  angles  in  one  direction, 
equal  to  the  product  of  the  sines  in  the 
opposite  direction. 

Take  H  as  the  vertex,  and  WPB  as 
the  base  ;  then, 

sin  HWP.  sin  HPB  .  sin  HBW 

=  sin  HBP.  sin  HPW.  sin  HWB.  ; 

that  is,  sin  (12)  .  sin  (8)  .  sin  (6)  =  sin  (10)  .  sin  (4)  .  sin  (2),  as  was 
otherwise  obtained.     The  angles  at  the  point  used  as  the  ver- 

12 


1/8  GEODETIC  OPERATIONS. 

tex  are  not  involved  in  this  equation,  so  they  must  be  involved 
in  a  station  adjustment,  or  in  a  triangle  condition. 

If  one  should  find  it  difficult  to  conceive  a  pyramid  con- 
structed in  this  way,  he  can  without  trouble  secure  the  side 
equation  in  the  manner  made  use  of  on  page  174,  in  which  we 
started  from  HW  '.  sin  (2)  =  HB  .  sin  (6). 

In  the  next  equation  obtain  a  value  of  HB.  in  another  tri- 
angle, as  HB  .  sin  (10)  =  HP.  sin  (8)  ;  then  in  terms  of  HP.,  as 
HP.  sin  4=ffW.  sin  12. 

This  is  as  far  as  we  can  go,  as  we  have  returned  to  the  start- 
ing-point. Suppose  we  start  from  WP. 

WP.sin(ii)  =  W#.sin(7); 

WSsin  (6)  =  Wtf.sin(g); 

WH.sm(i)  =  WP.sin(4)'t 

by  multiplying,  sin  (i  i)  .  sin  (6)  .  sin  (i)  =  sin  (7)  .  sin  (9)  .  sin  (4). 
The  same  can  be  obtained  by  taking  Was  the  vertex,  and 
BHP  as  the  base,  the  angles  in  one  direction  will  give 

sin  WPH.  sin  WHB.  sin  W£P=s'm  WBH.  sin  WHP.  sin  WPB. 

In  writing  down  the  equations  to  be  used,  a  good  plan  is  to 
put  down  the  sides  emanating  from  the  pole  to  all  the  other 
points,  putting  the  line  first  in  the  first  member,  and  then  in 
the  second  ;  as, 


coming  back  to  the  first  line  used.     Then  we  put  in  the  angle 
that  is  opposite  the  side  in  the  other  term  ;  as,  (i  i)  opposite 


CALCULATION  OF  THE    TRIANGULATION.  179 

WB,  (7)  opposite  WP,  in  accordance  with  the  trigonometric 
theorem. 

The  following  rule,  so  frequently  quoted,  is  taken  from 
Schott  (C.  S.  Report,  1854). 

The  only  choice  in  selecting  the  station  to  be  used  as  the  ver- 
tex, or  pole,  as  it  is  sometimes  called,  is  to  take  that  vertex  at 
which  the  triangles  meet  which  form  the  triangle  equations  of 
condition,  and  to  avoid  small  angles,  since  the  tabular  differ- 
ences, being  large,  will  give  unwieldy  coefficients.  It  is  some- 
times difficult  to  determine  the  precise  number  of  condition 
equations  that  can  be  formed. 

The  least  number  of  lines  necessary  to  form  a  closed  figure 
by  connecting/  points  is/,  and  gives  one  angular  condition. 
Every  additional  line,  which  must  necessarily  have  been  ob- 
served in  both  directions,  furnishes  a  condition  ;  hence  a  sys- 
tem of  /  lines  between  /  points,  I—  p-\-\  angle  equations, 
where  it  must  be  borne  in  mind  that  each  of  the  /  lines  must 
have  both  a  forward  and  a  backward  sight. 

When,  in  any  system,  the  first  two  points  are  determined  in 
reference  to  one  another  by  the  measurement  of  the  line  join- 
ing, then  the  determination  of  the  position  of  any  additional 
station  requires  two  sides,  or  necessarily  two  directions;  hence 
in  any  system  of  triangles  between/  points,  we  have  to  deter- 
mine /  —  2  points,  which  require  2(p  —  2)  directions,  or  by 
adding  the  first  2p  —  3.  Consequently,  in  a  system  of  /  lines, 
/ —  (2p  —  3),  or  /—  2/  -\-  3  sides  are  supernumerary,  and  give 
an  equal  number  of  side  equations. 

We  have,  therefore, 

/  —  /  -j~  I  angle  equations ; 
/  —  2/>  -|-  3  side  equations ; 
2/—  3^  +  4  in  all. 

It  is  apparent  that  each  point  may  be  taken  as  the  pole,  and 


1 80  GEODETIC  OPERATIONS. 

as  many  side  equations  formed  as  there  are  vertices.  In  a 
quadrilateral,  for  instance,  if  four  side  equations  are  formed,  the 
fourth  equation  would  involve  the  identical  corrections  con- 
tained in  the  others.  Since  there  are  only  12  angles  in  all, 
these  can  be  incorporated  in  two  equations,  each  of  which  con- 
tains 6  angle  corrections. 

From  the  formulas  just  given,  it  will  be  seen  that  4  condi- 
tional equations  will  be  sufficient  in  a  quadrilateral ;  I  side 
and  3  angle  equations,  or  2  side  and  2  angle  equations,  but 
never  more  than  2  side  equations. 

The  method  of  station  adjustment  differs  somewhat  from 
the  foregoing  when  the  values  of  the  angles  depend  upon  di- 
rections. 

In  nearly  all  refined  geodetic  work  angles  are  so  determined  ; 
that  is,  the  zero  of  the  circle  is  set  at  any  position,  the  tele- 
scope is  pointed  upon  the  first  signal  to  the  left,  and  the  mi- 
crometers or  verniers  read ;  the  telescope  is  then  pointed  to 
each  in  succession  and  the  readings  recorded.  After  reading 
the  circle  at  the  last  pointing,  this  signal  is  again  bisected  and 
readings  made,  likewise  with  the  others  in  the  reverse  order. 
The  telescope  is  reversed  in  its  Y's  and  a  similar  forward  and 
backward  set  of  pointings  and  readings  made.  These  form  a 
set.  The  circle  is  then  shifted  into  a  new  position  and  another 
set  observed,  as  already  described.  The  average  of  the  direct 
and  reversed  readings  of  each  series  is  taken  as  a  single  deter- 
mination of  a  direction. 

—  Let  x  be  the  angle   between   the 

zero  of  the  instrument  and  the  direc- 
tion   of  the  first  line,  A,  B,  C,  etc., 
lg  the  angles  the  other  lines  make  with 
the  first,  whose  most  probable  val- 
FIG  ues  are  to  be   determined,  and  let 

/«„  «,*,  m*  .  .  .  be  the  reading  of  the 
circle  when  pointing  to  the  signals  in  order,  of  which  x^  is  the 


CALCULATION  OF   THE    TRIANGULAT10N.  l8l 

most  probable,  and  the  errors  of  observation  ml  —  x^.  Suppos- 
ing no  errors  existed,  we  should  have  the  following  equations: 


tl  —  x^  —  A  =  o  ; 
*  —  x^  —  C  —  o. 


The  second  series  would  give 


mi  —  xi  —  °  I  m*  —  xi  —  A  =  o  ; 

m*  —  x^  —  B  =  o  ;  mt3  —  x^  —  C  =  o  ; 

and  the  wth,       mn  —  xn  =  o  ;  win—  xn  —  A  =  o  ; 

vi,  i  —  xn  —  B  =  o  ;  win  —  xn  —  C  =•  o. 


The  most  probable  values  will  be  those  the  sum  of  the 
squares  of  whose  errors  is  a  minimum.  Also,  the  errors 
squared  must  be  multiplied  by  the  corresponding  weights,/,, 
P\>P\  -  '  •  Pv  P*  •  •  •  which  will  give 


-  *,  -  ^)a  +  Af«  -  *.  - 


etc.,  etc. 

Differentiating  with  respect  to  x^  xv  xz  .  .  .  A,  B,  C  .  .  .  and 
placing  the  differential  coefficients  separately  equal  to  zero,  we 
shall  have 


182 


GEODETIC  OPERATIONS. 


V;/i  + AX1  + AX2  +  AX"  •  •  • 
=  (A  +A1  +Aa  +  A'  •  •  •  X+A1 


A'«,  +  AX1  +AX2  +AX3  •  •  • 

=  (A  +A1  +A2+A3  •  •  •  K+A'^+A'^+A3^  •  -  •  ;  [  (A) 

A*».+AX'  +AX2  +A3^s3  •  •  • 

=  (A  +  A1  +  A9  +A3  •  •  •  ;*.+A' 
etc.,  etc.; 


A'^i'  H  A1^1  +  AX1  +  •  •  • 

-  (A1  +  A1  +A'  •  •  •  M  +  AX  +AX  +AX  . .  - 

AX1  +  AX1  +AX1  +  •  •  • 

=  (A2  +  A2  +A2  •  •  •  )&  +/.X  +AX  -f  AX  . . . 

'  =  (A3  +Aa  +A3  ••  -  )<^+AX  +AX  +AX . 


In  these  equations  x^  —  mlt  x^  —  mv  x^  —  m3  .  .  .  are  the 
errors  of  observation  ;  calling  these  ;r,,  xv  x^  .  .  .  they  will  rep- 
resent the  corrections  of  the  first,  second,  third  .  .  .  pointings 
from  the  zero-mark — usually  a  small  quantity. 

By  multiplying  out  the  parenthesis  in  the  second  member  of 
(A),  and  transposing  all  the  terms  from  the  first,  we  have 

o  =  A*,- A^+AX-AX'+AX- AX'+AX- AX' 


Introduce  into  each  parenthesis  #/,  —  ;«„  except  the  first, 


JO.  .  .  ', 


CALCULATION  OF   THE    TRIANGULATION.  183 


For  .*•,  —  T#,  substitute  ^r,,  and  for  aw,1  —  ;;/,  write  ;«,';  re- 
membering that  mts  ,  which  is  to  take  the  place  of  mf  —  me,  does 
not  mean  the  /th  reading  on  the  sth  arc,  as  recorded,  but  the 
recorded  reading  minus  the  reading  of  the  zero  on  that  arc. 

This  will  reduce  the  last  equation  to 

o  =  A-*'i  +AX  —  AX'  +  AX  —  AX'+AX  —  AX* 


AX'  +  AX'+AX 
=  (A  +A1  +A2 


In  the  same  manner  the  other  equations  (A)  reduce  to 

AX'+AX'+AX'--. 

-  (A  +  A1  +  A2  +  A3  •  •  •  K+  A 

AX'+AX'+AX3-.. 

-  (A  +  A1  +  A2  +  A3  •  •  •  K+ 


Likewise,  equations  (B)  reduce  to 

AX'+AX'+AX1-  •• 

=  (A1  +  A1  +A1  "-)A  +  AX  +AX  +  AX  •  • 

AX2  +  AX3  +  AX2--- 

-  (A'+A2  +A2  ..'.)*  +AX  +AX  +AX  •  . 

AX3+AX3+AX3.«. 

=  (A1  +A3  +  A3  •  -  •  )C  +  AX  +  AX  +  AX  •  • 


(D) 


When  the  signals  observed  upon  are  numerous,  the  solution 
of  equations  (C)  and  (D)  would  be  very  laborious. 


1 84  GEODETIC  OPERATIONS. 

Captain  Yollond,  of  the  Ordnance  Survey  of  Great  Britain, 
found  the  method  of  successive  approximations  sufficiently 
accurate. 

Suppose  x^  x^  ^3  ...  severally  equal  to  zero  in  (D),  from 
which  we  find  the  first  approximation  : 

A,  _  AX'  +  AX1  +  AX1  • .  • . 
_  AX3  +  AX*  +  AX'  •  •  • . 

•£»      —   ~~       ~T5      I       T~5      I       Ta 


_  AX8  +  AX8  +  AX8  •  •  • 
A3+A3+A3..- 

Substituting  these  values  in  (C),  we  obtain  a  new  value  for  x^ 

_p*(m*—A')-\-p?(m?—  B')  +  p*(m*—  C)  .  .  . 
**  A+A1 


A  +  A'+A'  +  A3... 


Substituting  these  values  in  (D),  we  obtain  the  second  approxi- 
mation, or 


A'+A'+A1.-. 


CALCULATION  OF   THE    TRIANGULATION.  1  8$ 

_  /.'«  -  *.)  +  A2«  -  *.)  +  A'«  -*.)..-. 

A'+A'+A1.-. 

™  _  A8«  -  *.)  +  A'«  -  *.)  +  A'«  -  *.)  •  •  • 

''1 


The  values  can  be  further  substituted  in  (C)  and  the  result- 
ing values  of  x^  xv  x^  .  .  .  placed  in  (D)  for  the  third  approxi- 
mation for  A,  13,  C  .  .  .  However,  the  second  has  been  found 
sufficient  in  good  work. 

The  weights  for  observed  directions  is  unity,  and  zero  for 
any  directions  that  could  not  be  observed.  The  work  can  be 
materially  shortened  by  pointing  on  the  first  object  on  the  left, 
as  the  beginning  of  each  series;  and  in  each  successive  series 
the  readings  of  the  first  direction  should  be  diminished  by  the 
preceding  direction,  in  this  way  taking  as  a  zero  the  first  direc- 
tion of  each  series. 

In  the  ordnance  survey,  the  readings  on  the  initial  object 
were  made  the  same  in  the  different  series  by  adding  to  the 
average  readings  of  the  microscopes  on  each  signal  such  a 
quantity,  positive  or  negative,  as  to  make  the  initial  readings 
the  same. 

Considering  the  weights  unity, 

m*  +  m,1  -f-  m,1  .  .  . 

si    =  -  , 


where  n  represents  the  number  of  series,  or  A'  =  the  arithmet- 
ical mean,  say  M^  in  the  same  way  we  find 

B'  =  MV        C  =  Mt... 

Substituting  these  values  in  the  expressions  for  xv  x^  ,  .  .  we 
have 


1 86  GEODETIC  OPERATIONS. 


,  =  ~«  -M,+  m?  -  M,  +  m?  -  M9  .  .  .  ), 


and  similarly  for  x^  xt  ...  we  get  —  jfcf,1,  —  M3l  .  .  . 
Placing  these  values  in  the  second  approximation, 


A"  =  -«-  MS  +  mJ-MJ  +  mS-M,1  .  .  .) 


B"  =  l-W- M;  +  <-  M;  +  <  -M; . . . ); 

C"  =  l-(m?  -  Ml  +  <  -  ^,'  +  '«ss  -«'—> 


We  have  first  obtained  a  constant  reading  for  the  initial  di- 
rection, either  its  angular  distance  from  an  azimuth-mark,  of 
by  making  the  first  direction  zero.  We  then  found  the  aver- 
age of  each  direction,  giving  A'  =  M^  B'  —  M^  ...  or  the 
arithmetical  mean  as  the  first  approximation.  Next  we  sub- 
tracted each  average  from  each  reading,  giving  a  set  of  errors 
— the  average  of  those  in  the  same  series  giving  M*,  M*  .  .  . 

Afterwards  these  are  taken  from  the  readings  of  the  corre- 
sponding series,  giving  diminished  values  of  each  direction  ; 
and  the  average  of  these  diminished  directions  gives  the  second 
approximation. 

A  symbolic  analysis  can  be  seen  in  the  appended  table,  fol- 
lowed by  an  example  taken  from  the  Report  of  the  Ordnance 
Survey,  1858,  page  65: 


CALCULATION  OF   THE    TRIANGULATION. 


I87 


Initial  Object. 

A. 

B. 

C. 

Averages. 

m 
m 
m 

ml 
ml 

ml 
ml 
ml 

ml 
ml 

Average  

Mi 

Mt 

M3 

m   1           M 

ml  -  M, 

Ml 

ml  -  Mi 

ml  —  M, 

ml  -  M» 

M3' 
Ml 

ml  —  Ml 
ml  -  Ml 
ml  —  Ml 

Ml  -  Ml 
ml  -  Ml 
ml  -  Ml 

ml  -  Ml 
ml  -  Ml 
ml  -  Ml 

Averages...  . 

A" 

B" 

C" 

No.  of 
Series. 

Initial  O. 

A  -  ii*  7'. 

B  =  37°  34'- 

C  =  97"  54'- 

D   =   220°  3'. 

Average 
Errors. 

2 

4°2l'29".2I 
29   .21 

36'  '.04 
35    91 

I4".07 

47"-84 

19".  oo 

18    18 

3 

29   .21 

34  -2I 

ir  .86 

4 
5 
6 

29    .21 
29   .21 
2Q     21 

32  4i 

10  .71 
ii  .91 

46  .05 

48  .30 

16  .30 
14  .17 
18    *^Q 

Average 

29".  21 

34"-  64 

12".  14 

47"-4° 

i7"-25 

Errors. 

oo".oo 

,OO 

+  i"-40 

•4*-    I      27 

+  i"-93 

4-  o".44 

+  i"-75 

+    O      Q7 

-f-  i".io 

.00 

.00 

-  o  .43 

-    2   .23 

—  o  .28 
-  i  -43 

-  i  -35 

-  o  .95 
—  3    08 

—  o  .23 
-  I    .19 

oo 

+    1     34 

4-  o    67 

28".  1  1 
28  48 

34"-  94 
35    18 

I2".97 

46".  74 

17".  90 
17    45 

29    44 

12     OQ 

30  .40 
29  .81 
28    <u 

33  -60 

II    .90 

12    .51 

46  .28 

48  .90 

17  .49 

M  -77 

, 

Average 

29".  13 

34"-54 

I2".37 

47"-3i 

I7".ii 

1  88  GEODETIC  OPERATIONS. 

This  gives  the  directions  as  follows  : 

Initial  object  =  4°  2i/29//.i3  ; 
direction  A"  —  II  7  34  .54; 
direction  B"  —  37  34  12  .37; 
direction  C"  =  97  54  47  .31; 
.direction  D"  =  220  3  17  .11. 

The  third  approximation,  obtained  in  the  same  way,  gave. 
omitting  degrees  and  minutes:  initial  object  =  29//.i2,  A'"  — 
34".55,  B"'  —  12".  40,  C"  =  47"-34,  D"'  =  I7".o8,  values  dif- 
fering from  the  above  in  the  hundredths  place  only. 

The  angles  depending  upon  these  directions  will  be  involved 
in  the  figure-adjustment,  so  their  corrected  values  should  be 
written  A  +  (i),  B  +  (2),  £+(3)  ...  in  which  (i),  (2),  (3)  ... 
are  the  corrections  obtained  in  the  figure-adjustment.  In  this 
operation  the  directions  obtained  at  different  stations  have  not 
the  same  weight  ;  however,  this  can  be  computed  from  the 

formula  already  given  on  page  125,  where  we  found  p  =  •  —  ^ 
i  n* 


n* 

So  we  find  the  residuals  by  taking  the  difference  between 
the  individual  diminished  measures  and  the  average,  and  di- 
vide the  number  of  readings  on  that  direction  squared  by 
twice  the  sum  of  the  squares  of  the  residuals  ;  in  the  case  of 

16 
= 


To  illustrate  the   formation  of  the  equations  of  geometric 
condition  let  us  take  an  example. 
The  ajigles  adjusted  at  stations  are  : 


CALCULATION  OF  THE    TRIANGULATION. 


I89 


at  T,  M  =    oo°oo'oo" 

F=    83  30  34  -866  + (i) 

W  —  287  14  13  .822 +  (2) 
at  M,  T  —    oo  oo  oo 

£F=    66  56  10  .619 +  (4) 

F=  293  57  16  .395+  (6) 
at  F,   W  =    oo  oo  oo 

J/  =    20  oo  09  .436  +  (7) 

T=  349  33  27  -528  + (i i) 
at  W,  F  —    oo  oo  oo 

7-  =     13  17  05  .983 -f  (12) 
M=  332  59  01  -843 +(15). 

In  the  triangle  MTF,  we  are  to  find  the  angles  at  each  ver- 
tex, as  follows  : 

83°3o'34".S66+(i)  =  MTF; 

66    243  .605— (6)  =FMT,  or  360°— direction 

Ftrom  T; 

302641   .9o8+(7)-(n) =  MFT\ 

i8o°oo'oo'/.379+(i)-(6)+(7)— (i  i)=  sum  ; 

1800000  .015  =  1 80°  +  spherical  excess ; 

o  =  o".364  +  (i)  -  (6)  +  (7)  -  (i  i).  Equation  (I) 

To  find  MFT,  we  subtract  the  direction  of  T  from  W,  from 
360°;  this  gives  angle  WFT;  to  this  add  the  direction  of  M 
from  W,  or  the  angle  WFM. 

In  the  triangle  TMW, 

72°45/46//.i78-(2)  =  MTW\ 

66  56  10  .6i9+(4)  =  WMT\ 

40  18   4  .I40+(i2)— (15)  =MWT; 

i8o°oo'oo".937-(2)+(4)+(i2)-(i5)=  sum  ; 

1 80  oo  oo  .01 1  =  i8o°+spherical  excess  ; 

0  =  +0//.926-(2)+(4)+(i2)-(i5).  Equation  (II) 


igO  GEODETIC  OPERATIONS. 

In  the  triangle  WTF, 

I3°I7'05".983+(I2)  =  TWF; 

156  1621    .044+(i)-(2)  =  WTF\ 

102632  .472  — (11)  =  TFW; 

i79°59'59"499+(0-(2)-(II)+(12)  =  sum  '> 

1800000  .009  =  1 80° -[-spherical  excess; 

o=  -o".5io  +  (i)-(2)-(n)+(i2).  Equation  (III) 

In  the  quadrilateral  TMFW,  the  side  equation  is 

_  sin  TMW.  sin  FWT .  sin  TFM 
~  sin  MWT.  sin  TFW.  sin  FMT' 

sin  TMW  =    9.9638207,6  +      8.965(4)  (8.965  =  tab.  dif.)  ; 

sin  FWT  =    9.3613403,1+    89.174(12); 

sin  TFM  =    9.7047600,1  +    35-824[(7)  —  (n)] ; 

29.0299210,8. 


=    9.8107734,2  -f    24.826[(i2)  -  (15)]  ; 
sin  TFW  =    9.2582687,7—  114.245(11); 
sin  FMT  =    9.9608833,6  -      9.354(6)  ; 
29.0299255,5. 


o  =  -  44-7  +  8.965(4)  +  9.354(6)  +  35-824(7)  +  78.421(1 1) 

4- 64. 348(1 2) +  24.826(1 5).  Equation  (IV) 

These  four  equations  are  solved  for  the  unknowns,  which 
are  applied  to  the  given  directions  with  their  proper  signs,  or 
to  the  angles  directly,  as  just  deduced. 


CALCULATION  OF   THE    TRIANGULATION.  19! 

In  an  extended  triangulation,  the  position  of  every  point  is 
influenced  to  a  certain  extent  by  the  directions  at  the  adjacent 
signals  ;  consequently,  it  is  advisable  to  include  in  the  equations 
of  condition  as  many  directions  as  possible.  The  influence  of 
these  directions  upon  an  initial  point  diminishes  with  the  dis- 
tance, and  finally  becomes  inappreciable,  so  that  the  triangula- 
tion can  be  divided  into  segments,  each  containing  a  conven- 
ient number  of  conditional  equations.  The  corrections  of  the 
first  are  computed,  and,  as  far  as  they  go,  these  corrected  val- 
ues are  substituted  in  the  equations  of  condition  in  the  second 
figure,  and  the  sum  of  the  squares  of  the  remaining  errors,  each 
multiplied  by  its  corresponding  weight,  made  a  minimum. 

The  equations  of  condition  (I),  (II),  (III),  (IV)  .  .  .  may  be 
written 


o  =  a  -f  a,x, 
o  =  b  +  b,x, 
o  =  c  +  c^  +  cs.  •  .  .  ;  h  •  •  •  •  (E) 


If  /,,/„  ...  be  the  weights,  corresponding  to  the  corrections 
#,,.*:.,...,  the  requirement  that  the  sum  of  the  squares  of  the 
errors  be  a  minimum  is 

A*i*  +  A*.*  +A*i*  •  •  •  =  a  minimum.      .    .     (F) 

Differentiating  (E)  and  (F),  we  have 


o  =  a,dxv  -f  a,d 
o  =  b,dx,  -f-  bji 
o  =  c,dx,  +  cjx*  +  ctdxt  .  .  . 


o  = 


192  GEODETIC  OPERATIONS. 

Solving  these  equations  as  explained  on  page  160,  we  have 


.   .   .    (G) 


Substituting  the  values  of  x»  xv  x^  ...  as  found  in  these  equa- 
tions in  (E),  we  have 


or,  o  = 


In  the  same  way,  remembering  that  (a2)  is  the  sum  of  the 
squares  of  quantities  like  a,  as  a?-{-  a?-\-  a* .  .  .  and  (ab)  =  a161 


•     (H) 


/„  /„/,...,  being  auxiliary  multipliers,  have  their  values  ob- 
tained from  (H)  and  substituted  in  (G),  giving  the  numerical 
values  of  x^,  x»  x^  .  .  . 

Instead  of  using  fltft  .  .  .  the  Roman  numerals  I,  II,  III  .  .  . 
will  be  found  more  convenient,  especially  when  the  conditional 
equations  are  so  numbered.  The  normal  equations  can  be 
more  readily  formed. 

To  illustrate,  suppose  we  have  the  following  equations  of 
condition : 


CALCULATION  OF   THE    TRIANGULATION.  193 

I,  O  -  -  I.4042-(2)  +  (5)  -  (7)  +  (8)  ; 

II,  o  =  -  2.7737-(2)  +  (4) ; 

III,  o=-o.959S-(8)  +  (n); 

IV,  o=  -  1.2157— (3) +  (4); 

V,  o  =  -  o.92o4-(3)  +  (5)  -  (7)  +  (10)  ; 
VI,  o  =- 0.8424-0)  +  (4); 
VII,o=-o.320i-0)  +  (5)-(7)  +  (9);  I    m 

vin, o  = +0.999  -(O +  (3); 


XXV,  o=  + 

-6.188(7); 
etc.,  etc. 


(i),  (2),  (3)  ...  represent  the  corrections  to  directions  of  the 
same  number  ;  then  we  multiply  the  terms  involving  (i),  (2), 
by  the  reciprocals  of  their  weights,  giving 

(i)  =  —  o.oSooVI  —  o.oSooVII  —  o.oSooVIII 

((i)  occurs  in  VI,  VII,  and  VIII,  and  0.800  is  the  reciprocal  of 

its  weight)  ; 


(2)  =—  0.2060!  —  o.2o6oll  —  0.0000309XXV  ; 
(3)=-o.i58oIV-o.i58oV  + 


(4)  ^+o.33SoII+o.33SoIV+o.338oVI+5.263302XXV  ; 

(5)  =+0.2260!  +o.226oV  +  o.226oVII-3.5i922XXV  ; 

etc.,  etc. 


(K) 


These  values  of  (i),  (2),  (3)  ...  are  substituted  in  the  equa- 
tions of  condition  (I),  giving  numerical  values  for  I,  II,  III  .  .  . ; 
13 


194  GEODETIC  OPERATIONS. 

then  these  values  substituted  in  equations  (K)  give  the  val- 
ues of  the  corrections  (i),  (2),  (3)  .  .  .  ,  which,  when  applied  to 
the  directions,  will  give  their  most  probable  values,  satisfying 
the  geometric  conditions. 

For  the  various  methods  of  adjustments,  see  : 

Jordan,  Handbuch  der  Vermessungskunde,  vol.  i.,  pp.  339- 

346- 

Bessel,  Gradmessung  in  Ostpreussen,  pp.  52-205. 

Clarke,  Geodesy,  pp.  216-243. 

Wright,  Treatise  on  the  Adjustments  of  Observations,  pp. 
250-348. 

Ordnance  Survey,  Account  of  Principal  Triangulation,  pp. 
354-416. 

C.  and  G.  Survey  Report  for  1854,  pp.  63-95. 

Die  Konigliche  Preussische  Landes-Triangulation,  I.,  II., 
and  III.  Theile. 

When  a  number  of  normal  equations  are  to  be  solved,  it  is 
found,  by  some,  desirable  to  eliminate  by  means  of  logarithms  ; 
but,  as  logarithms  are  never  exact,  there  will  always  remain 
small  residuals  when  the  corrections  are  applied.  Direct  elimi- 
nation is  preferable,  unless  the  coefficients  are  large ;  then  the 
logarithmic  plan  is  somewhat  shorter.  We  will  illustrate  with 
an  algebraic  equation  : 

3«  -f-    x  -\-  2y  —    2—22  =  0;.     .     .     .  (i) 

4,r  _    y  _j_  3(gr  -  35  =  0;.     .     .     .  (2) 

4*  +  ix  -  2.y  -19  =  0;.     .     .     .  (3) 

2U  +  4 y  -f  22  —  46  =  O (4) 

If  the  first  equation  were  multiplied  by  ^,  the  coefficient  of 
u  would  be  the  same  as  in  (3),  and  upon  subtraction  the  u's 
would  disappear.  To  multiply  by  ^  is  simply  adding  log  4  — 
log  3  to  the  logarithms  of  the  coefficients  of  (i),  omitting  32* ; 
we  write,  then,  the  logs  of  these  coefficients : 


CALCULATION  OF   THE    TRIANGULATION. 


195 


Dog  of  coef.,  o.oooo 
log  4  —  log  3,  0.1248 
add  0.1248 

nat.  numbers,   1.333 
coef.  of  (2),       3 
subtract        —  1.667 


y- 

z. 

22. 

0.3010 

no.oooo 

n  i.  3424  = 

o; 

0.1248 

0.1248 

0.1248  ; 

0.4258 

#.1248 

n  i.  4672  ; 

2.666 

-  1-333 

-  29.33  ; 

2 

+ 

-  19   ; 

+  4.666 

-  1-333 

-  10.33.  • 

(5) 

Take  a  factor  that  will  make  the  coefficient  of  u  in  another 
equation  equal  to  its  coefficient  in  one  of  the  other  equations, 
multiply  (4)  by  2,  or  add  to  the  logs  of  the  coefficient  in  (4), 
the  log  of  2  =  0.3010. 


Logs  of  coef.  of  (4),  .... 

log  2,  

add 

nat.  numbers,          .... 
coef.  of  (3),  3—2 

subtract  —  3 


y- 

z. 

46. 

0.6020 

O.3OIO 

#1.6627; 

.03010 

.030IO 

.03010  ; 

0.9030 

O.6O2O 

7*1.9637  ; 

8 

4 

-92           ; 

2 

-  19       ; 

0 

4 

-73-    -    - 

(6) 


Continue  to  eliminate  the  same  quantity  from  all  the  remain- 
ing equations  until  one  equation  remains  with  one  unknown 
quantity. 

The  only  advantage  that  this  method  suggests  is,  that  only 
one  quantity  is  used  as  a  multiplier  to  make  the  coefficients 
identical ;  that  factor  is  usually  a  fraction,  whose  log  is  simply 
the  difference  between  the  logs  of  the  numerator  and  denom- 
inator. 

Mr.  Doolittle,  of  the  Coast  Survey,  has  developed  another 
method  of  elimination,  which  can  be  found  in  the  Report  for 
1878,  page  115. 


I96 


GEODETIC  OPERATIONS. 


REDUCTION   TO  CENTRE   OF   STATION. 

the  directions  adjusted  it  is  necessary,  when  an  eccen- 
tric position  has  been  occupied,  to  reduce  the  corrected  ob- 
served dire<£jons  to  their  equivalents  at  the  centre,  before 
cornputing  tJje  distances  and  co-ordinates, 


FIG.  25. 

0  =  observed  angle; 
x  =  desired  angle. 

Angle  from  signal  to  A  =  a,  to  B  =  b, 
Angle  m—A-\-x  —  0-}-B. 


sin  A  — 


CALCULATION  OF   THE    TRIANGULATION. 


197 


As  B  and  A  are  always  very  small,  they  may  be  regarded  as 


equal  to  B  sm  I  ",  and  ^4  sin  I  ", 


r  sin  (0  _|_  »)  r  sin  n 

—  '  —  —  - 


hence, 
Also, 


rsin(0-f-»)        rs'mn 
b  sin  i"          #  sin  i"" 


angle  between  B  and  C  =  /2  OC  -f- 


r .  sin  (fl-f-0-1-07)       r  sin  «  . 


.  sm  i 


a  sin  i' 


From  the  above  equations  it  will  be  seen  that  all  angles  that 
are  read  from  the  same  initial  point  have  for  their  corrections 
the  same  last  term  ;  so  this  term  can  be 
computed  for  each  initial  direction  and 
applied  to  the  various  angles.  In  both 
terms  of  the  corrections  there  are  two  con- 
stants  for  each  station,  rand  sin  i";  so  the 
work  can  be  facilitated  by  tabulating  their 
values.  The  signs  of  the  terms  will  de- 
pend upon  the  sign  of  the  sine  function. 

It  will  assist  in  the  computation  to  take 
the  angle  between  the  signal  and  the  first 
point  to  the  right  and  continue  in  that 
direction. 

Signal  23  feet  from  instrument.  Angle 
between  //"and  C.T  =  71°. 


C.T 


Fzc.  26. 


Log  dist.  //"to  C.Tin  M.  =  4-7534757; 
log  dist.  H  to  B.K  in  M.  =  4.6503172  ; 
log  dist.  //"to  H.K\\\  M.  =  4.8385482; 
log  dist.  //"to  £7  in  M.  =  4.6145537. 


198 


GEODETIC  OPERATIONS. 


Sig.  toC.2'  + 
C.Tlo  B.K. 

Sig.  tocr+ 

C.TloH.K. 

Sig.  to  C.T+ 
C.TloC. 

Sig.  to  J3.K  + 
B.K  to  H.K. 

SIWfSc+ 

Direction  .... 

107°  24'  I?" 
9.9796466 
0.8457389 
5.3496828 
5.3I4425I 
1.4894934 
30".  86 

120°  53'  55" 
9.9335264 
0.8457389 
5.1614518 
5-3I4425I 
1.2551442 
17".  99 

166°  06'  57" 
9.3801384 
0.8457389 
5.3854463 
5.3i4425t 
0.9257487 
8".  42 

120°  53'  59" 

9.9335214 
0.8457389 
5.1614518 
5.3144251 
I-255I372 

17".  99 

166°  06'  55" 
9-3801555 
0.8457389 
5.3854463 
5.3144251 
0.9257658 
8".  43 

Log  r(M).  .  .  . 
Co.  log  dist.  . 
Co.  log  sin  i" 

Cor  

Signal  to  C.  T. 

Signal  to  B.K. 

•jl   —  OO  —  OO 

107  —  24  —  17 

9  9756701 

Log  ?(M)  

0.8457389 

0.8457389 

Co.  log  dist  
Co.  log  sin  i"  

Cor  

5.2465243 
5.3144251 
1.3823584 

24".  12 

5.3496828 
5.3144251 
1.4894934 

30".  87 

CORRECTED   ANGLES. 

C.T.to  B.K=  36°  24'  i7"+3o".86-24".i2=36°  24'  23//.74; 
C.T.ioH.K=  49°  53'  55"+i7".99-24".i2=49°  53'  48//.8;  ; 
C.T  to  C.  =  95°  06'  57"+  8".42-24//.i2=95°  06'  4i".3o; 
=  13°  29'  42//-i-i7/'.99-3o//.87=i3°  29'  29//.i2  ; 
C.  =  58°  42'  38"+  8//.42-3o//.87=s8  °42'  i5x/.55. 

The  distances  used  above  were  obtained  from  the  observed 
values  of  the  angles,  and  are,  therefore,  only  approximate.  In 
the  case  of  refined  work,  it  will  be  necessary  to  use  these  cor- 
rected values  and  again  compute  the  distances  ;  then,  with  the 
correct  distances,  recompute  the  reduction  to  centre. 

With  the  most  probable  value  for  all  the  directions,  the 
angles  of  all  the  triangles  can  be  found  by  taking  a  given  di- 
rection from  360°,  or  by  adding  or  subtracting  two  or  more 
directions. 


CALCULATION  OF   THE    TRIANGULATION.  199 

Then  with  a  base,  measured,  or  previously  computed,  each 
side  can  be  found  by  the  trigonometric  formulae  a  = 

b  sin  (A   -  -)  b.  sin  (c  -  -] 

^— ,  and  c  = 5_,    in   which   *   is   the 

sin  (B-'-)  sin  (B-'-) 

computed  spherical  excess,  as  obtained  from  using  approxi- 
mate lengths  and  angles. 

If  there  are  more  than  one  base  in  the  triangulation-net,  the 
most  satisfactory  method  is  to  compute  each  base  from  all  the 
others,  and  take  the  mean  of  the  logarithmic  values  so  found; 
or,  if  the  entire  scheme  is  involved  in  a  single  figure,  the  abso- 
lute term  in  the  side  equation  can  be  made  equal  to  the  ratio 

D 

of  the  two  bases,  -~,  instead  of  unity. 

We  can  also  find  the  length  of  any  line  as  influenced  by  two 
or  more  bases.  Let  B^  Bv  B3  ...  be  the  bases,  x  the  most 
probable  value  of  any  side  in  the  triangulation,  and  rlf  rv  r3 
...  be  the  ratio  of  each  side  respectively  to  x\  the  errors 
then  will  be 


Now,  if  A'  A>  A  •  •  •  be  the  weights  of  the  bases,  then 

p,(r,x  -  B$+pfrs  -  B$  +  A(r,*  -  £,)'  •  •  •  =  a  minimum. 


Placing  the  differential  coefficient  with  respect  to  x  =  o,  we 
find 


200 


GEODETIC  OPERATIONS. 


From  this  we  can  find  the  most  probable  length  of  one  base 
from  all  the  others.  To  do  this  we  suppose  x  to  be  one  of 
the  bases,  say  B»  then  r,  =  I, 


A+A'.'+A'.' 
the  correction  will  be  x  —  Blt  or 


" 


A*.  +  A* 


A 


A 


The  adjustments  so  far  considered  affect  the  geometric  con- 
ditions, and  in  their  operations  may,  by  changing  the  direc- 
tions of  the  lines,  change  the  azimuth,  making  a  greater  or  less 
difference  between  the  observed  and  computed  azimuths.  In 
refined  geodetic  work,  the  azimuth  is  observed  at  least  twice 
in  each  figure,  and  sometimes  twice  in  each  quadrilateral. 


FIG.  27. 

Using  Wright's  figure  and  notation,  we  take  PQ&nd  TU  as 


CALCULATION  OF   THE    TRIANGULATION.  2OI 

two  lines  whose  azimuths  have  been  observed  with  the  simplest 
and  most  approved  connections.    PQ,  as  a  known  line,  enables 
one  to   compute  PR,  and  from  PR  we  can  go  direct  to  SR, 
thence  to  ST;  so  these  lines  are  called  sides  of  continuation. 
Let  Av  AV  A3  .  .  .  be  the  angles  opposite  the  sides  of  con- 
tinuation ; 
2?,,  Bv  J5a  .  .  .  the   angles    opposite  the  sides  taken  as 

bases ; 

Cv  Cv  C3  .  .  .    the  angles  opposite  those  sides  not  used ; 
Z»  Zv  the  measured  azimuths  of  PQ  and   777, 

supposed  to  be  correct,  and  therefore 
subject  to  no  change  ; 

Z'  the  computed  azimuth  of  TU,  reckoning  from  the 
south  around  by  the  west.  £7,,  £72,  C3,  Ct,  are  the  only  angles 
that  enter  into  this  computation  ;  and  the  excess,  £,  of  the 
observed  over  the  computed  azimuth  gives 


=  £,     .     .     Eq.(L) 


in  which  (£7,),  (Cy}  .  .  .  represent  the  corrections  to  £",,  £!,... 
Now,  since  the  triangles  have  had  their  angles  adjusted  to  the 
conditions  imposed  upon  them,  their  total  corrections  must  be 


.    .    .    .    (M) 

;)=o.j 

Also,  the  sum  of  the  errors  squared 

(A^f  -f- (•#))"  H~  (£1)'  •  •  •  —  a  minimum. 


2O2  GEODETIC  OPERATIONS. 

The  solution  of  these  equations  would  give 

At  =      IE,        A,  =  -££.. 


If  there  were  n  intervening  triangles,  we  would  find 


'A,-      —E,        A,  =  ---£-, 
2n    '  2n 


From  which  the  following  rule  is  deduced : 

"  Divide  the  excess  of  the  observed  over  the  computed  azi- 
muth by  the  number  of  triangles,  and  apply  one  half  of  this 
quantity  to  each  of  the  angles  adjacent  to  the  unused  side,  and 
the  total  quantity  with  its  sign  changed  to  the  third  angle.  In 
each  following  triangle  the  signs  are  reversed." 

The  discrepancies  between  the  observed  and  computed  lati- 
tudes and  longitudes  are  very  slight,  and  can  be  adjusted  arbi- 
trarily. 


GEODETIC  LATITUDES,   LONGITUDES  AND  AZIMUTHS. 


CHAPTER  VII. 


FORMULA  FOR  THE  COMPUTATION  OF  GEODETIC  LATITUDES, 
LONGITUDES  AND  AZIMUTHS. 

WHEN  we  know  the  geographical  position  of  a  point,  and 
the  distance  and  direction  to  another,  the  co-ordinates  of  the 
second  can  be  computed  from  the  data  just  named,  by  using 
formulae  for  the  difference  in  the  latitudes,  longitudes  and  azi- 
muths. 


In  the  above  meridian  section,  let  A  be  a  point  whose 
latitude  is  L.  By  definition  it  is  equal  to  the  angle  AN£, 
formed  by  the  normal  A N  and  the  equatorial  radius  EC. 

a*  —  tf 
AG  =  N,  e1  = 1 — ,  in  which  a  is  the  semi-major  axis, 

and  b  the  semi-minor. 


2O4  GEODETIC  OPERATIONS. 

r* 

The  subnormal  in  an  ellipse  MN '=  CM '.-,. 


AM  =  NM .  tan  L  =  CM.  -3 .  tan  L, 
a 


squaring,      AM*  =  CM*  .  -4  tan8  L.  (i) 

The  equation  of  an  ellipse  gives 
b\  . 


therefore  CM' .  ~t  tan4  L  =  -t(a*  —  CM9), 


CM*.-,  tana  L  =  a*-  CM*  \ 

clearing  of  fractions  and  transposing, 

CM\V  tan'  L  +  a*)  =  a4 ; 

hence  CM*  - 


tan3  Z -fa" 


sin*  L  „ 
substituting  - — -^j  for  tan*  L, 


GEODETIC  LATITUDES,   LONGITUDES  AND  AZIMUTHS.      2O5 

-^j-^a  a*  cos*  Z  a*  cos8  L 

~  ^s'm'L-}-alcosrL  =  b\i  —  cos2  Z) +V  cos8  £ 

a:4  cos2  Z 


of=    ,      a'cos£ 


From  definition  «V2  =  «2  —  ^2, 

«'  cos  Z-  02  cos 


ai  -  sn 

a1  cos  Z  #  cos  L 


4/0"  _  0V  sin2  L      <*Vi  —  (?  sin2  Z-        ^l  —  e*  sin2  Z' 
which  is  the  radius  of  a  meridian. 

AO          AO  a 

In  the  triangle  A  GO,  AG  —  -. — JT=T^  = /  —  —  = 

sin  A  GO      cos£       ^i—e'sirfL 

=  N,  or  the  normal  produced  to  the  minor  axis. 

The  ordinate  AM  can  be  found  from  the  equation  of  the 
ellipse,  a1 .  AM*  +  &  .  ~CM*  =  c?b\ 


<?V  -  VCM*      a*V          V  a3  cos'L 

AM  = r 


a*  ~    a*        a\i  —  e*  sin2  Z) 

F  cos8  L       _  b\i  -  sin8Z) 

"  i  -*8sin8  L~        ~  i  -*2sin2£ 

y  _  ^y  sin2  L-F  +  F  sin3  L 
i  —  e*  sin2  L 


2C»6  GEODETIC  OPERATIONS. 

-  ^  sin'Z(i-g')  _  a*  sin*  Z(i 


i  —  e*  sin3  L  i  —  e*  sin' 

therefore 

a(i  —  <?7)  sin  L 


AM  = 


Vi  -  i  sin'  L 


To  find   the   normal  AN,  we  take  the  triangle  AMN,  in 
which 


sm  I/I—  ^  sin8 


The  radius  of  curvature,  R  = 


dx* 


d*y  V  (dy\      Vx* 

In  the  general  equation,   -7-5-  =  --  =-,,  [•r~l   =  -r-«» 

dx*  ay  \dx'        ay 

substituting  these  values, 

-  L1  +    J     gy  w 


+  </    _  gy  w  +  y^i«  _  c^y  +  wy 

P  '    V  L        a  J    "  a4? 


In  this  expression  we  place  for  x  the  value  we  found  for  CM, 
and  for y  that  of  AM;  this  gives 


GEODETIC  LATITUDES,    LONGITUDES  AND  AZIMUTHS.      2O? 

pz'(i   —  e*y  sin"  L       a*(i  —  e*)*  —  a"(i  —  e*)*  sin* 
i  —  e*  sin2  L  I  —  e"  sin2  L 


R_       «(i-O 

(i  —  /  sin2  Z)5* 

The  terminal  points  «,  b,  c,  d  and  ^  of  the  radii  of  curva- 
ture form  an  evolute  ;  at  the  equator, 


At  the  pole, 

7_nno 

~ 


The  above  formulae  are  in  terms  of  geographic  latitude; 
the  geocentric  latitude  is  equal  to  the  angle  formed  at  the  cen- 
tre by  the  equator  and  radius.  In  the  figure  it  is  the  angle 
A  CM.  Calling  it  6,  we  have 

_  AM  _  a(\  —  e*)  sin  L  a  cosL 

=  MC  =    i^TUSTZ:"*  "*"    i  -Ssm'L 


b\ 

--  =  (  i  —  e]  tan  L  =  -.  tan  L. 
L  a 


208 


GEODETIC  OPERATIONS. 


It  is  always  less  than  the  geographic  latitude,  the  difference 
being  greatest  at  those  places  where  a*  —  F  is  the  greatest,  or 
at  latitude  45°,  N.  or  S.,  where  the 
difference  is  about  11'  30". 

In  the  adjoining  figure,  Pis  the 
pole,  E  the  plane  of  the  equator, 
A  and  B  two  points  on  the  earth's 
surface  whose  latitudes  are  L  and 
L',  co-latitudes  A  and  A',  and  the 
geodesic  line  AB  is  /.  An  and 
Bn',-t\\Q  normals,  are  N  and  N', 
and  R  and  R  the  radii  of  curva- 
ture. 

The  azimuth  is  estimated  from 
the  south  around  by  the  west;  the 
angle  PAB  =  180°  —  Z,  will  be 
designated  x,  and  the  angle  be- 
tween the  two  meridians  AP  and 
BPis  the  difference  of  longitude, 
dM. 

In  the  spherical  triangle  APB, 


FIG.  29. 


cos  A'  =  cos  A  cos  l-\-  sin  A  sin  /  cos  x. 

In  relation  to  A  and  A',  /  is  very  small,  so  that  a  series  involv- 
ing /  will  converge,  so  we  write  A'  =/(A  -j-  /). 
Developing  this  by  Taylor's  formula,  we  have 


1 '          1     I 

A.     —  /v  —I -jrt 

'     dl 


2.3.<#s 


In  order  to  find  these  differential  coefficients,  some  relation 
must  be  established  between  A,  A-f-  d\  and  dl.     Taking  these 


GEODETIC  LATITUDES,   LONGITUDES  AND  AZIMUTHS.      2OO, 

as  sides  of  a  differential  spherical  triangle  having  the  angle  x 
between  the  first  and  third  sides,  we  have 


cos  (A  -J-  dK)  =  cos  d/cos  A  -j-  sin  A  sin  <//cos  x. 

Obtaining  the  differential  coefficients  of  A  with  respect  to  /, 
we  substitute  them  in  (i) ;  this  is  accomplished  by  expanding 
the  last  equation  so  as  to  have 

cos  A  cos  d A  —  sin  A .  sin  d\=.  cos  dl  cos  A 

-j-  sin  A  sin  dl  cos  x ; 
cos  d\  =  I, 

and  sin  </A  =  d\.     This  reduces  the  equation  to 

cos  A  —  sin  A  d"k.  =  cos  A  dl-\-  sin  A  d/cos  x. 

Cos  A  may  be  assumed  equal  to  cos  \.dl,  and  therefore 
they  eliminate  each  other,  leaving  —jj  =  —  cos  x. 

</'A  <TA  dx  dx 

—77-  =  sin  x .  dx      — Tn  =  sin  x  -77  ;  but  -77  =  —  sin  x .  cot  A ; 
dl  dl  dl  dl 

-775-  =  sin'  x  cot  A,          also  —777-  =  sin*  x  cos  x(\  +  3  cot1  A). 
dl  dl 

Substituting  L'  and  L  for  A'  and  A,  and  remembering  that  cot 
A  =r  tan  Z.,  we  obtain  from  (i), 

L'—L  =  -I  cos  x  —  i/8  sin1  x  tan  L 

+  |/'  sin1  x  cos  x(\  +  3  tan1  L), 
14 


2IO  GEODETIC  OPERATIONS. 

x  =  1 80°  —  Z,         cos  x  =  —  cos  .2,        sin  x  =  sin  Z, 
L'  -  L  =  -  ^/Z 


-  |/3 sin2  ZcosZ(i  +  3  tan'Z); 

or,  -*TZ:=/cosZ+i/2sin'ZtanZ 

-  &3  sin2ZcGsZ(l  +  3  tan'Z). 


The   value  of  /  has  been  considered  as  expressed  in  arc, 

while  in  computation  it  will  be  given  in  linear  measure.  There- 

>y- 

fore  /  =  -vv,  where  K  is  the  length  of  the  line,  and  N  the  ra- 
dius of  the  imaginary  sphere  on  which  L  is  a  point 


/fcosZ    ,  ^2sin'ZtanZ      JTsin'ZcosZ 
__  +_  ____..  ___ 


This  needs  a  further  transformation,  to  refer  the  formula  to 
an  ideal  sphere  whose  radius  is  the  radius  of  curvature  of  the 
middle  meridian.  This,  however,  cannot  be  known  until  L'  is 
computed ;  however,  we  can  start  with  the  value  of  R  for  the 
initial  latitude,  and  apply  a  correction.  The  reduction  is  made 

N 
by  multiplying  by  the  ratio  of  -77;  we  also  divide  by  arc  i"  to 

convert  the  arc  dL  into  a  linear  multiple  of  i' '.     This  gives 


rd77  sin° Zcos  z( 


GEODETIC  LATITUDES,   LONGITUDES  AND  AZIMUTHS.      211 
Denoting  the  radius  of  curvature  of  the  mean  meridian  by 

n  _    n 

Rf>t,  dL  must  be  increased  by  —  ~  —  -  .  dL\ 


~  (i  -  ?  sin2  Zji      (i  -  e1  sin'  Lm)* 

(i  -  e*  sin*  £„)•  -(i-e*  sin9 
'     I  -  *'  sin1  Z*(i  -  ^  sina  L 


Expanding  by  binomial  formula, 


-  *  sn    »   =  i  -     sn    ^      ^  sn    « 

—  /  sin2  Zi  =  I  -    *1  sin2  4       4 


subtracting  =  f^-2  sin2  L  —  f^2  sin2  Lm  .  .  .  , 

omitting  higher  powers  of  e\  or,  |<?2(sin2£  —  sin2Zw). 
This  can  be  reduced  as  follows: 

sin  (L  —  Z.OT)  —  sin  L  cos  Lm  —  cos  L  sin  Lm  ; 

sin  (L  -\-  Lm)  =  sin  L  cos  Lm  +  cos  L  sin  Lm  ; 

sin  (L  —  Lm)  sin  (L-\-Lm)  =  sin2  Z  cos3  Lm  —  cos2  £  sin2  Lm 
=  sin2Z(i-sin2Zw,)-(i-s 
=  sin2  L  —  sin2  Lm. 

Lm  is  the  mean  latitude  between  L  and  L  -f-  dL, 


sin  (Z  -  Lm)  sin  (Z  +  Zw)  =  sin  (L  -  L  -  \dL)  sin  (2L  +  \dL) 
=  dL  sin  Z,  .  cos  Z,  nearly. 


212  GEODETIC  OPERATIONS. 

R~R,n 


Then' 


_  3  f    a(i  —  e*)dL  .  sin  L  .  cos  L        (i  —  e1  sin2  Lm)* 
~%e  (i-S  sin*L)i.(i  ~ 

dL.s'mL.  cos  Z, 


As  this  is  a  small  quantity,  it  can  be  converted  into  a  linear 
function  by  dividing  it  by  arc  i",  giving 


R  -  R,n       _  dL*  sin  Z  .  cos  L 

~ 


-  ^3  sin3  £)*  arc 
Introducing  this  into  (2),  we  have,  after  placing 


n-          l  r-         tanZ 

"7"  ~//' 


/f  ^3  sin  L  cos  £ 

~  x/  *  C 


R  .  arc  ix/  *  ~  (I  -  S  sin2  Z)«  arc  ix/> 

-flX  =  ^Tcos  Z.B      K*  sin2  Z.  C-  /WT  sin2Z. 


The  last  term  was  devised  by  Professor  Hilgard,  in  1846. 

The  factors  B,  C,  D,  E,  are  given  in  the  last  pages  com- 
puted for  Clarke's  (1866)  Spheroid. 

When  the  line  is  not  more  than  fifteen  miles,  the  third  term 
can  be  omitted,  and  7*2  put  for  (dL)*,  giving  as  an  abbreviated 
formula 

-  dL  =  KcosZ.  B+  K*sm*Z. 


Francoeur  has  given  a  purely  trigonometric  method  for  de- 
riving the  formula  just  obtained. 


GEODETIC  LATITUDES,   LONGITUDES  AND  AZIMUTHS.     21$ 

Using  the  same  figure,  we  write  PA  =  90°  —  Z,  PB  =  90° 
-L'. 

In  the  spherical  triangle  PAS,  we  know  PA,  AB  =  !,  and 
the  angle  PAB  =  180°  —  Z. 

cos  PB  —  cos  PA  cos  /-)-  sin  PA  sin  /cos  PAS; 
cos  (90°  —  Z7)  =  cos  (90°  —  L)  cos  / 

+  sin  (90°  —  Z)  sin  /cos  (180°  —  Z)  ; 
sin  L'  =  sin  L  cos  /  —  cos  L  sin  /  cos  /?. 

Subtracting  both  sides  from  sin  L, 

sin  L  —  sin  L'  —  sin  Z  —  sin  Z  cos  /  -f-  cos  L  sin  /  cos  Z 
=  sin  Z(  i  —  cos  /)  -f-  cos  L  sin  /cos  Z 
=  2  sin  L  sin4  £/  -f-  cos  L  sin  /  cos  Z. 

sin  Z  -  sin  L'  =  2  sin  £(Z  —  L'}  cos  |(Z  +  L')  ; 
suppose  Z  -  L  =  d,     then     Z  -f  L'  =  2Z-(Z-Z')  =2L-d, 
and          \(L  -  L'}  =  K        ^(2  L-d)  =  L-  \d, 


therefore  sin  L  —  sin  Z'  =  2  sin£d?cos(Z  —  £</) 

=  2  sin    <fcos  Z  cos  £<3?--sin  L  sin 


that  is,  2  sin  ^  cos  \d  cos  Z  -{-  2  sin2  ^  sin  Z 

=  2  sin  Z  sin2  £/  -f-  cos  Z  sin  /  cos  Z. 

Dividing  by  2  cos  L  cos2^,  we  obtain 

sin  \d         sin2  £*/sin  L         sin  Z  sin2  ^/         cos  Z  sin  IcosZ 
cos    </  '     cos"    */  cos  Z  ~  cos  Z  cos2  %d  ~""     2  cos  Z  cos" 


214  GEODETIC  OPERATIONS. 

tan  L  sin2   l        sin/cos^ 


tan 


tan  \d(l  +  tan  £*/tan  L)  = ^-y-^tan  L  sina  \l-\-^,  sin  /  cos 


Placing  H  for  the  last  parenthesis,  we  may  write 


TT 

tan  \d(\  +  tan  ^  tan  L)  =   -  5-=--^  =r  #(i  -f  tan2 


tan  ^  +  tan4  \d  tan  Z  =  H-\-ff  tan3  ^  ; 

tan|^+(tanZ  -H)tan*$d=H.      .     .     .     (i) 
In  the  expression  for  H,  /is  small  ;  so  we  can  write  for  sin  / 

r  r 

its  serial  value,  sin  /  =  /  —  ^-,  also  sin2  %/  =  —  ,  which  will 
give 

H  =  i/cos  Z+  \r  tan  L  -  Ty  '  cos  Z. 

We  must  now  solve  (i)   for  \d\  for  short  we  will  put  \d  = 
c,  and  tan  L  —  H  =  h,  so  (i)  reduces  to 

TJ 

tan  c  +  tana  c  .  h  =  H,         or         tan  c  =  —  r—  -,  --  .    (2) 

I  -)-  h  .  tan  c     ^  ' 

Neglecting  h  .  tan  c,  we  have  tan  c  =  H,  as  the  value  of  first  ap- 
proximation. Substituting  this  value  for  tan  c  in  the  second 
member  of  (2), 

TT 

tan  c  =  ,     ?r  =  H  —  /iff',  by  division. 


GEODETIC  LATITUDES,   LONGITUDES  AND  AZIMUTHS,      21$ 

Again,  substituting  this  second  approximation  in  (i),  we  have 
as  the  third  approximation 

tan  c  =  H-  hH*  +  2/?H3  ; 
by  continuing  to  the  fourth,  we  get 


The  development  of  an  arc  in  terms  of  its  tangent  gives 

c  =  tan  c  —  \  tan3  c  -\-  ^  tan6  c  .  .  . 
Placing  in  this  the  value  of  tan  c,  just  found,  we  have 


Resuming  our  notation,  we  have  c  =  \d,  and  h  =  tan  L 
—  H\  this  gives 


h  =  tanZ,  —  |/cosZ—  ^/*  tan  L  +  TV/'cosZ; 
i^  =  i/cos  Z+  i/2  tan  Z  -  -jV/3  cos  Z 

-  (tan  L-  i/cos  Z-  £/'  tan  Z+  TV/3  cosZ.  .  .  ) 
(|/  cos  Z+  i/2  tan  L  —  Ty  3  cos  Z)3  .  .  . 

Multiplying  this  out  and  retaining  terms  of  /  to  /3, 


=•!/  cos  Z+i/'  tan  Z  —  £/"  tan  Zcos'Z-  ^r  cos  Z 

+  TV/3  cos3  Z—  i/3  tan5  £  cos  Z-f  i/3  tan'Z  cos3  Z 
=  |/cosZ+i/atanZ(l  -  cos3Z)  -1V^3cosZ(i  -  cos'Z) 
-i/3cosZtanaZ(i  —  cosaZ) 
=  i/cos  Z  +  i/a  tan  L  sin"  Z  -  T^/  3  cos  Zsin'  Z 

-i/'cosZsin'Ztan'Z 
=  i/cos  Z+  i/J  tan  L  sins  Z-  ^3  cos  Zsin'Z(i+3  tan'Z)  ; 


2l6  GEODETIC  OPERATIONS. 

ar^/cosZ-f-i/'tanZsin'Z  —  £/3cosZsin'Z(i  -f  3tan'.£). 

Remembering  that  d  =  L  —  Lr,  we  have  here  the  identical 
formula  given  on  page  210. 

There  is  still  another  form  to  which  this  can  be  reduced,  in- 
volving more  factors  that  can  be  tabulated,  and  at  the  same 
time  occurring  in  the  computation  of  longitude  and  azimuth. 

K  K(i  -/sin'Z)* 

Take  (2)  and  substitute  u"  for  -^^  -  Tl  —  —  *  -  r  —  r/  -  -  ; 

N  sin  l"  tfsm  \" 

multiply  this  by  -~-,  it  becomes  -  —  —  -5  —  ;  reducing  this  frac- 
tion and  omitting  terms  above  ^*, 


=  j  _[_  /  -  f  sin8  L  =  i  +  e\i  -  sin8  Z)  =  I  +  ^  cosa  L. 
The  first  term  becomes 


likewise  the  second  term 

•-=(!+**  cos'  L)  (u"  sin  Z)a  tan  L  ^-  ; 
substituting,  it  becomes 
L'=L-(i+e*  cosa  L)u"  cos  Z—  ^  sin  I'^i+^cos'  L)  (u"  sin  Z)9. 


This  formula  gives  good  results  for  distances  of  twenty  miles 
and  under. 

The  algebraic  sign  of  the  different  terms  depends  upon  the 
trigonometric  functions  of  Z. 


GEODETIC  LATITUDES,   LONGITUDES  AND  AZIMUTHS.      21? 

Whenever  the  sides  are  more  than  a  hundred  miles  long,  this 
method  of  difference  of  latitude  will  introduce  some  errors. 

In  that  case,  the  method  to  be  followed  is  to  solve  the 
spheroidal  triangle  formed  by  the  two  points  and  the  pole,  in- 
volving as  trigonometric  functions  the  sought  co-latitude,  azi- 
muth and  difference  of  longitude. 


LONGITUDE. 

Referring  to  the  figure  on  page  208,  and  using  the  same  no- 
tation, we  get  in  the  triangle  ABP,  sin  A'  ;  sin  x  : :  sin  /  :  sin  dM. 

Supposing   the  radius  of  the  sphere  to  be  Bn'  =  N,  I  =.  -^, 
and  that  /and  dM  are  proportional  to  their  sines 

A"  1 

dM=-=— 


A'  and  L'  being  complementary,  sin  A'  =  cosL'. 

If  very  accurate  geodetic  computations  are  to  be  made,  a 
small  correction  must  be  applied,  owing  to  the  difference  be- 
tween the  arcs  and  sines  of  small  angles.  This  correction  can 
be  taken  from  the  table  on  page  274. 

The  quantity  dM  increases  towards  the  west.  The  alge- 
braic sign  of  the  equation  depends  upon  sin  Z,  which  is  -f-  be- 
tween o°  and  180°. 


If  we  place   u"  =  •„   . 77,        dM  = j- 

cos  L 


K       K" 

this,  however,  supposes  that  -^  =  ^=,  which  is  only  approxi- 
mately so. 


. 
218  GEODETIC  OPERATIONS. 


AZIMUTH. 

The  initial  azimuth  of  the  base  or  initial  line  being  known, 
that  of  any  line  emanating  from  either  extremity  can  be  known 
by  adding  to  or  subtracting  from  the  azimuth  of  this  base  the 
angle  between  the  two  lines. 

Let  A  B  be  the  base,  and  C  a  point  making  angle  m  with 
AD  at  A:  if  the  azimuth  of  AB  =  Z,  that  of  AC  will  equal 
Z  ±.  m.  But  in  order  to  determine  the  direction  of  a  line  ex- 
tending from  C,  as  CD,  the  azimuth  of  CA  must  be  known. 
If  the  earth's  surface  were  a  plane,  Z'  would  equal  180°  -f-  Z  ; 
but  the  spheroidal  shape  of  the  earth  complicates  this  as  well 
as  all  other  geodetic  problems. 

Again  referring  to  figure  on  page  208,  in  the  triangle  APB, 
by  Napier's  Analogies, 

tan  \dM  :  cot  \(x  +  x'}  :  :  cos  £(A/  —  A)  :  cos  i(A  -f  A'), 


from  which  cot^O  +  x')  =  C°S        ,       >    tan  \dM  '; 

COS   "$\A*    —  "•) 

but  x=iSo°-Z, 

therefore     x  -f  x'  =  180  -  Z-f  x'  =  180°  +  (x1  —  Z)  ; 

cot  i[(i8o°  +  (x'  —  Z)~\  =  —  tan  \(x'  —  Z), 
but  (x'  —  Z}  —  dZ,    also    \  =  90°  —  L,    and    ^'=go°  —  L', 

therefore  A  +  A7  =  180°  -  L  -  L'  =  180°  -  (L  +  L'}  ; 

A7  -  A  =  90°  -  L'  -  (go0  -  L}  =  L  -  L'  =  dL  ; 

cos  ![(i8o°  -  (L  +  L'}}  =  sin  \(L  +  //)  -  sin 
the  formula  then  reduces  to 


GEODETIC  LATITUDES,    LONGITUDES  AND  AZIMUTHS.      2 19 

tan  \dZ  =  -  tan 


COS 

Supposing  that    tan  \dZ  :  tan  \dM  : :  dZ  :  dM, 

j-r       jut-  sm  A* 
—  dZ  =  dM 


This  is  not  exactly  correct;  the  correction  can  readily  be  found 
by  adding  a  term,  say  x,  to  the  fourth  term  of  the  above  pro- 
portion and  solving  for  the  value  of  x.  It  will  be  found  that 
3  cos*  Lm  sinZ.,,,  sin2  i"  must  be  added  to  the  above  value 


cos*  Lm  sin  Lm  sin2  i" 
of  dZ.  The  factor  ----    can   be   tabulated  as 

factor  F,  a  table  of  which  is  appended  ;  the  expression  then 
becomes 


-  dZ  = 


i 

cos    dL 


The  algebraic  sign  of  dZ  will  depend  upon  dM.  As  the  azi- 
muth is  estimated  by  common  consent  from  the  south  around 
by  the  west,  so  long  as  the  initial  azimuth  is  less  than  180°, 
the  reverse  azimuth  Z'  =  Z  +  180°  +  dZ  ';  but  if  more  than 
1  80°,  Z'  =  Z  -  180°  -f  dZ. 

A  table  of  values  is  given  for  cos  \dL  for  lines  of  twenty 
miles  and  under.  The  term  involving  f  can  be  omitted,  and 
the  value  of  dM  deduced  above  substituted  in  its  place,  giving 

Z  —  Z  ±  1  80°  -j  --  jj-  .  sin  Lm.     It  has  also  been  found  suf- 

ficiently accurate  to  omit  cos  \dL  and  write  —  dZ  —  dM  sin  Lm. 
In  accurate  work  the  azimuth  should  be  determined  at  least 
once  in  every  figure  by  astronomic  observation.      This  opera- 
tion is  fully  described  in  works  on  practical  astronomy. 


220  GEODETIC  OPERATIONS. 


L.  M.  Z.  FORM    FOR   PRIMARY  TRIANGULATION. 


Z 

L 
7, 
& 

/.' 

Mount  Blue  to  Mount  Pleasant 
Mount  Pleasant  and  Ragged  (. 
Mount  Blue  to  Ragged  (360°  - 

26       19 
85       35 
300        44 
+              50 
301        34 
121        34 

£* 

i-34 
3-7i 
5-05 
5-os 

\.  is  to  the  left  of  M.  P.)  

(85«  -  35'  etc.  -  26°  -  19'  .  .  .  )  

Ragged  to  Mount  Blue  (Z  +  d 

Z—  180")  

L 
dL 
L1 

44 
44 

43 
3° 

40.121 

5S-978 
44-M3 

Mount  B 
i  10740.  6  A 
Ragged. 

ue 

i 

M 

iM- 
M' 

70 
69 

20          11.921 
II          27.659 
08          44.262 

/,  log  5.  0443070  

K 
cos  Z 
B 

h 

5.0443070 
9.7084622 
8.5.104895 

A-» 
sin8  Z 
C 

3d  term 
4th  term 

K 
sinZ 

A> 
cos£' 
ar.  co. 

dM 

10.08861 
9-86854 
1.39991 

(dL)* 
D 

6-5372 
2-3933 

h 

K*  sin" 
£ 

3.2632 
9-9571 
6.2069 

3.2632587 

1.35706 

8.9305 

9.4271 

ist  term 
2d  term 

dL 
3d  and 
3  4th 

-dL 

Lm 

1833.406 
22-754 

0.085 

-  0.267 

5.0443070 
9  9342721" 
8.5090158 
0.1446254 

arg. 
K 

dM 
cos 

+  317 
+    99 

(dM)* 

dM 

sin  Lm 
cos  \dL 
ar.  co. 

—  dZ 

2d  term 

10.896" 
7.840 

1856.160 
-     .182 

8.736 
3-6322302 
9.8454305 
0.0000040 

1855-978 
44-28-12.13 

99 
3.6322302 
—  4287.757 

3.4776647" 
3003.76 
-      0S   3003.71 

Notes  upon  the  Computation. — The  angle  Mount  Pleasant 
and  Ragged  is  recorded  minus,  since  the  second  point  is  to  the 
right  of  the  first — contrary  to  the  graduation  of  the  instrument. 
1 80°  is  subtracted,  since  the  general  direction  is  east.  In  the 
sixth  column,  —  218  and  317  correspond  to  the  correction  due 
to  the  supposition  that  the  arc  and  sine  are  equal  ;  the  value, 
99,  is  added  to  dM ;  dM\s  negative,  since  sin  Z  is  minus.  In 
the  azimuth-computation,  the  second  term,  .05,  is  the  antiloga- 
rithm  of  8.736 ;  this  is  negative,  therefore  .05  is  subtracted. 

The  data  here  used  were  taken  from  the  Coast  Survey  rec- 
ords. There  the  Z,  M  and  Z  were  computed,  using  Bessel's 
constants:  the  results  are,  ZB  =  O5".55,  Zc  —  $".0$,  LB 
=  43"-955.  Lc  =  44"i43,  M»  =  43".578,  M?  =  ^".262. 

In  using  the  abbreviated  formula,  the  third  and  fourth  terms 


GEODETIC  LATITUDES,  LONGITUDES  AND  AZIMUTHS.    221 


would  be  omitted  in  latitude,  but  in  their  place  should  be  in- 
serted VD. 

In  longitude,  the  correction  for  the  ratio  of  sine  to  arc  is  not 
inserted.  Also,  for  azimuth-computation  cos  \dL,  and  (dM^F 
are  insignificant,  and  consequently  left  out.  The  terms  that 
are  disregarded  could  not  affect  the  result  beyond  the  tenth 
of  a  second,  in  lines  less  than  a  hundred  miles  in  length.  A 
very  convenient  form  in  use  in  the  U.  S.  Geological  Survey  is 
appended,  employing  the  abbreviated  formula  already  given  : 


Names  of 
Stations. 

Position. 

Observed 

Angles. 

Correc- 
tion by 
L.S. 

Correc- 
tions 
arbi- 
trary. 

Spheri- 
cal 
Angles. 

Spheri- 
cal 
Excess. 

Final  Plane 
Angles. 

Big  Knob  

Sought, 

'44  17  55-62 

" 

—  .4 

55-22 

-x.83 

'44  17  53-39 

Holston  

Right^H 

c    *z, 

III 

13  29  28.86 

" 

28.76 

-1.83 

13  29  26.93 

High  Knob..  .    Left,     £ 

22    12   41.69 

—  .18 

41.51 

-1.83 

22  12  39-68 

Comput- 
Letter. 

Logarithms 
of  their  Sines. 

Calculation  of  the  Sides. 

Sides  in 
Yards. 

Designation. 

5. 

9.7660909 

log  J?£  =  4.839*933 
a.  c.  log  sin  S  =  0.2339091 

4.8780609 

Holston—  High  Knob. 

R. 

log  LS  =  4.4409976 

safefc*h-  «•«-*-• 

log  sin  Z,  =  9.5775136 

4.4798652 

Big  Knob—  High  Knob. 

L. 

log/?.9  -  4.6506160 

4.6894836 

Holston—  Big  Knob. 

The  column  marked  correction  by  L.  S.  is  for  the  correc- 
tions obtained  in  figure-adjustment.  When  it  is  not  possible 
to  make  this  adjustment,  the  error,  after  deducting  spherical 
excess,  must  be  distributed  arbitrarily.  If  the  angles  are  ap- 
proximately equal,  one  third  the  error  should  be  applied  to 


222 


GEODETIC  OPERATIONS. 


each  angle  ;  if  not  equal,  the  distribution  should  be  propor- 
tional to  the  size  of  the  angles.  If  one  signal  should  be  dim, 
or  uncertain,  it  may  be  best  to  give  to  the  angle  between  it  and 
the  other  point  the  bulk  of  the  error.  Occasionally  the  angle 
deserving  the  greatest  correction  can  be  determined  by  ex- 
amining the  individual  readings.  If  they  vary  considerably, 
showing  a  wide  range,  the  inference  is  that  the  average  is 
somewhat  uncertain,  and  that  the  principal  source  of  error  in 
the  triangle  is  at  this  point.  Such  evidence  as  this,  and  the 
appearance  of  the  signals  from  each  other  should  have  some 
weight  in  distributing  the  error.  The  most  convenient  form 
of  blank  for  computation  is  to  have  three  or  four  sets  of  the 
upper  slip  printed  on  the  left  side,  and  the  same  number  of 
the  lower,  on  the  right  side  of  a  book. 


Names  of  Stations. 

LATITUDES. 

L'=  L-u"  (i-f  «"  cos2  L)  cos  Z  -  i  sin  i"  sin*  Z»"2(i-f  e»  cos"  L)  tan  L. 

Holston  

M 

£ 

1 

.M 

4       « 

£             M 

i  I 

§  4 

ii 

^      H 
High  Knob.  .  . 

Authority, 
Latitude  (L).  . 

log  K  (yds) 

U.  S.  Geo.  Survey. 
.....'=3627  27.41 

^  sin  i"  

.  .  —   4  3845448 

.  —   4  8780609 

IQO- 

2  log  sin  Z.  . 

10g^sini"" 

log«"  =    3-3484585 
log(i  +  «*  cos'2  /,)-.=    0.0018710 
log  cos  Z.  (—  )=    9.8460032 

.  .  —    6  69691701 

.  .=   0.0018710! 
..=    9.8685368 

log  tan  L  .  .  . 
log  2d  term  . 

2d  term  .... 
L-\-L'     . 

log  ist  term.  . 

—    3.  1963327 

.=    0.6576932 

•  •=      4"-55 

.  .  —  73  21  01.84 

2d  term  
SL 

..(-)=              4-55 

1567.02 
(-(-)  —      26  07  02 

L 

—  36  27  27  41 

Latitude  (L'). 

L  +  L' 

2 

.=364030.92 

=365334.43 

GEODETIC  LATITUDES,  LONGITUDES  AND  AZIMUTHS.    22$ 


LONGITUDES. 

AZIMUTHS. 

REMARKS. 

M' 

«»sinZ 

L+L> 

~*~    co&L'  ' 

Authority 
Longitude 

log  sin  Z. 
log  u"     . 

U.  S.  G.  S. 
M  =  82  04  38.17 

U.  S.  G.  S. 

134  32  39-47 

180 

Azimuth  Z  — 

Z  (-M  1  80°               — 

(+)  =     9-8529118 
—     3  3484585 

314  32  39.47 

.    Z  +  Z' 

log  sin  —  •  ...  = 

...(-L)  — 

log  cos  L 
log  (&W). 

SM  ... 

3.2013703 

9.7761771 
3.2984111 

(-)-)  _     9.9029592 

(+)  =     3-2984111 
<+)  =     1987.98 
•  •  •("{")  —       33  °7  98 

log  SZ  (-)  = 
<5Zin  seconds..  .  = 
SZ  (—  )  — 

3.0745882 
1187.38 

19  47.38 
314  32  39.47 

M  

—  82  04  38  17 

Z(+)i8o°  = 
Azimuth  Z'            — 

M' 

314  12  52.09 

The  latitude  blank  should  occupy  the  left,  and  the  longitude 
and  azimuth  the  right  side  of  a  book.  Two  forms  should  be 
on  each  page,  the  second  serving  as  a  check  computation,  by 
determining  the  third  point  of  the  triangle  from  the  other  end 
of  the  base.  For  example:  in  triangle  ABC,  suppose  L.  M. 
Z.  of  A  and  B  is  known,  C  can  be  determined  from  A,  and 
also  from  B.  The  average  of  these  values  is  to  be  taken,  to 
be  used  in  connection  with  A  or  Z?  in  determining  D,  etc.  .  .  . 


224  GEODETIC  OPERATIONS. 


CHAPTER  VIII. 

FIGURE   OF  THE   EARTH. 

WITH  the  geographical  positions  of  the  termini  of  a  line  and 
its  length  known,  it  is  possible  to  find  an  equivalent  for  its 
length  along  a  meridian  or  a  parallel,  thus  obtaining  a  value 
for  a  degree  in  that  latitude. 

Assuming  that  the  earth's  meridian  section  is  an  ellipse  of 
small  ellipticity,  we  can  develop  a  formula  giving  the  length  of 
an  arc  in  terms  of  the  terminal  latitudes,  the  semi-axes,  and 
ellipticity.  Also  the  problem  almost  the  converse,  by  which 
the  values  of  the  axes  and  ellipticity  can  be  found. 

Let  L,  L',  and  /  represent  the  terminal  and  middle  latitudes 
of  an  arc  whose  amplitude  is  /I  ;  a,  b,  and  f,  the  semi-major, 
semi-minor  axes  of  the  meridian-section,  and  the  ellipticity  ;  S, 
the  length  of  the  arc  ;  r,  the  radius  vector,  and  6,  the  geocen- 
tric latitude. 

The  equation  for  the  ellipse  is 

- 


x  =  r  cos  6,  y  —  r  sin  6,  substituting  in  (i), 

r>  cos"  6      r*  sin8  8  _ 
*  ~l> 


cos'  0    ,    sin"  8        I 
divide  by  r1,  --  +  —-  =  (2) 


FIGURE   OF  THE  EARTH. 

On  page  207  we  found 

tauten  A        or 
0 

f  ,  ~ 

from  which  sin*  6  = 


, 
os2  6       a'cos*L' 

b*  sin*  L  cos*  0 


. 
a  cos"  L 

Substituting  for  cos*  6,  i  —  sin'  6,  and  solving,  we  get 

•  t  ft  _  b*  sin*Z 

"Vcos'Z  +  ^sin'r 

By  a  similar  process  we  get 

t  a  _  _  dcos*  L 
~4' 


Placing  these  equivalents  in  (2), 


a*  cos4  L  -f-  64  sin4  L 


(3) 


In  the  ellipse,  tf  =  a*(i  —  ff  in  which  e  is  the  ellipticity  ;  sub- 
stituting this  in  (3), 

—  f)*sin*  L       I 


«4  cos4  L  -\-a\\  —  f)4  sin^L  ~  P' 

Dividing  out  ^",  and  writing  i  —  sin*  L  for  cos*  L,  after  reduc- 
tion, we  have 


=  a\i  —  4«  sin  *£  -f-  6e*  sin*  Z-  —  4^'  sin*  Z.  +  *4  sin*Z)  ; 

omitting  terms  involving  powers  of  f  above  the  second, 
15 


226  GEODETIC  OPERATIONS. 


r  =  a(i  —  £  sin8  L). 
The  formula  for  rectifying  a  polar  curve  is, 


ds  I    W       dr*  dr  .     _         _ 

-jyr    =r  A    /  r  --jfi  -f-  -Tr  a.  ~/T  =   —   2d:£  Sin  L  COS  A 

-rp  =  I  —  2£  -}-  4£  sin8  Z. 


This  is  obtained  by  differentiating  the  equation 

a"  tan  0  =  V  tan  Z,         or        tan  0  =  (i  —  £)"  tanZ; 

dZ 


j  —  £*  sin2 L  cos8  L 

—  a(i  —  2s  -\-  3«  sin"  Z.) ; 


omitting  in  the  above  all  terms  involving  £  above  the  second 
power  before  extracting  the  square  root. 

ds  =  a(i  —  2£+3£  sin2  L)dL  =  a(i  —  -  —  |-£  cos  2ZyZ, 

placing  sin8  Z  =  £(i  —  cos  2Z). 

Integrating  the  above  between  the  limits  Z  and  L',  we  have 
s  =  a[(i  —  |f)  (Z  -  Z')  -  f  £(sin  2Z  -  sin  2Z')], 

a  —  b 
b  —  «(i  —  £),  from  which  £  =  —  — .    Substituting  this  in  the 

above  equation, 


FIGURE   OF   THE  EARTH.  227 


=  (l±f)  (Z  -  i-)  - 


—  f(#  —  £)  (sin  Z  cos  L  —  sin  U  cos  Z').  .     (4) 


Z-Z'  =  A,        and 


sin  A.  =  sin  (Z-  —  Z')  —  sin  Z  cos  Z'  —  sin  Z'cosZ; 
cos  2/  =  cos  (Z  -j-  Z')  =  cos  Z  cos  Z'  —  sin  Z  sin  Z'; 
sin  A  cos  2/  =  sin  Z  cos  Z  cos2  L'  —  sin"  Z .  sin  Z'  cos  Z' 

—  sin  L  cos'Z  cos  Z'  +  sin"  L'  sin  Zcos  Z. 

Substituting  in  this 

sin*  L'  =  i  —  cos*  Z',        also        sina  Z  =  I  —  cos*  Z, 
it  reduces  to 

sin  A  cos  2/  =  sin  Z  cos  Z  —  sin  L  cos  Z'', 

which  is  the  same  as  the  last  term  in  (4) ;  therefore 

s  -  \(a  +  ff)l  -  f  (a  -  b]  sin  A  cos  2/.  (5) 

This  requires  a  particular  ellipsoid  from  which  to  obtain  the 
value  of  a  and  b,  but  it  gives  a  means  of  finding  a  and  b^'\i  all 
the  other  terms  are  known,  which  is  the  problem  geodesy  at- 


228 


GEODETIC  OPERATIONS. 


tempts  to  solve.     Suppose  s,  A,  /,  s',  A',  /'  be  the  lengths,  am- 
plitudes, and  mean  latitudes  of  two  arcs,  we  will  have 


s  =  \(a  -f-  £)A  —  |(0  —  b)  sin  A  cos  2/; 
s'  =  \(a  +  b)\'  -  \(a  -  b)  sin  A'  cos  2/'  ; 


(a 
solving  for  —  ±--,  and 


a  -f-  b       s'  sin  A  cos  2/  —  j  sin  A'  cos  2/' 
2      ~  A  sin  A'  cos  2/'  —  A'  sin  A  cos  2/ 


a  —  b 


s'\  —  sV 


3  '  A'  sin  A  cos  2/  —  A  sin  A'  cos  2/" 


from  which  a,  b,  and  f  can  be  found,  s  and  s'  are  the  distances 
between  parallels,  whereas  in  practice  our  lines  make  an  angle 
with  the  meridian,  so  that  its  projection  upon  the  meridian 
must  be  found. 

To  find  the  effect  of  errors  in  the  values  s  and  s'  upon  a 
and  b,  we  would  differentiate  the  above  equations,  regarding  3 
and  s'  only  as  variables.  In  the  result  the 
denominators  would  remain  ;  consequently 
the  minimum  error  would  occur  when  the 
denominator  is  a  maximum,  that  is,  when 
2/'  =  o,  and  /  =  90°,  or  when  one  arc  is  at 
the  equator  and  the  other  near  the  pole. 

Let  P  be  the  pole  of  the  spheroid,  PM 
and  PN  two  meridians  passing  through  the 
points  M  and  TV,  whose  geographic  and 
geocentric  latitudes  are  Z,,  L',  0,  and  0' '. 
PM  =  90°  -  0,  and  PN  =  90°  -  6',  from 
FlG-  3°-  which  NE  =  S—  6',  which  we  will  call  x ; 

also  the  line  NM=  s,  a  known  quantity. 


FIGURE   OF   THE  EARTH.  229 

In  the  spherical  triangle  MPN,  by  Gauss's  formulae, 


sin$(PN-  PM}  cos$MPN  =  sin  \MN  sin  \(PMN  -  PNM); 
cos  ±(PN—  PM  )  cos  \MPN  =  cos  pflV  sin  \(PMN+  PNM). 


Dividing  the  first  by  the  second, 


=  180°  -  Z,         PNM  =  Z  -  1  80 
hence          i(/>3flV  -  /WJ/)  =  i(36o°  -  (Z4-  Z7)) 


and  %(PMN+  PNM)  =  \(Z  -  Z). 

Substituting  these  values, 


sin^CZ+Z)         ,  ^  j 

Placing  h  =    ini/y__  ^-y  we  have  tan  j  =  *  tan  -  ;  writ- 
ing for  tan  -  and  tan  -  their  developments, 


Solving  this  equation   for  x  in  terms  of  s,  by  approximation, 


230  GEODETIC  OPERATIONS. 

we  have  for  the  first  value  x  —  sk ;  substituting  this  for  x*  and 
x',  we  have,  after  transposing, 


x_sh      sVi_sW      £^__£^!  _   sh     shtf  - 

~2   ~  ~2     '    24  ~~    24     '    240  ~~  240  '  '  2  ~*~  2  \        12 


for  the  second  approximation  ;  and  this  value  of  x,  substituted 
in  the  first  equation,  gives 


for  the  (A)  third  approximation. 


-  si 
_  s 


i(i  ~  cos  (Z'  -  Z)  -  j(i  -  cos  (Z+Z) 
sitf^Z'  —  Z) 

—  sin  Z  sin  Z' 


FIGURE   OF   THE  EARTH.  23! 

Regarding  the  earth's  meridian  section  as  an   ellipse,  we 
know  from  the  properties  of  an  ellipse  that 


x  =  a  cos  «,         and        y  =  b  sin  u, 

in  which  u  is  the  eccentric  angle,  or  reduced  latitude. 
Differentiating  the  above, 

—  dx  =  a  sin  udu,        dy  =  b  cos  udu, 

If  we  consider  this  point,  whose  co-ordinates  we  have  just 
written,  to  be  in  latitude  L,  and  an  element  of  the  elliptic 
curve  to  be  ds,  it  will  be  the  hypothenuse  of  a  right  triangle, 
in  which 

—  dx  =  ds  sin  L,        and        dy  =  ds  cos  Z., 
or     —  dx  —  a  sin  udu  —  ds  sin  L,       dy  —  b  cos  udu  =  ds  cos  L. 
Dividing, 


-r  tan  u  —  tan  Z-,        or        a  tan  u  =.  b  tan  Z,  (i) 


a  sn  w  ,  . 

and  (fr=___.  (2) 


The  value  found  for  *  in  (A)  was  for  a  spherical  surface  ;  to 
transform  to  an  ellipsoid  it  will  be  necessary  to  pass  to  a  dif- 


232  GEODETIC  OPERATIONS. 

ferential  triangle  on.  each.  In  the  figure  on  page  228,  suppose 
we  call  PNM  a  differential  triangle  on  an  ellipsoid,  in  which 
EN '=  dL,  NM  =  do-,  and  the  angle  PNM  —  a,  then  da  cos  a 
=  dL.  To  convert  dL  into  arc  measure,  we  multiply  it  by  the 
radius  of  curvature  of  the  meridian,  or 

ds  cos  a  =  RdL.  (3) 

Likewise,  if  we  conceive  the  same  triangle  to  be  on  a  sphere 
of  radius  a,  then  as  will  be  the  length  of  the  arc  MN,  then 

ado-  cos  OL  =  adu,         or        do-  cos  a  =  du.  (4) 


ds       RdL 
Dividing  (4)  by  (3),        -     =  - 


substituting  in  this  R  = 


also  from  (i),  we  find      sin  L  =  -( 5 ; 


ds 


_  f  sin'  Z*  '  I  -  ^  cos9 


f  _  iH^*L_  r  ' J 

L1  ~^i-^cosa«J 


—  e*  cos*  u 


i  —  e*  cos 


8  u  '  Fi  —  ^a  cos"  u—  e*  sina  « 

I   -  ?  COS8  jT~ 


FIGURE  OF  THE  EARTH.  233 

a(l  —  e')*  I 


—  e*  cosa  u  ' 


r   *-^  7 

Li  —  *•"  cos2  uJ 


or  =  «  ^i  -  <*  cos'  «,  (5) 


which  gives  the  relation  between  an  infinitesimal  length  on  a 
sphere  to  a  corresponding  length  on  an  ellipsoid. 

To  integrate  this,  Jordan  takes  a  spherical  triangle  with 
sides  equal  to  u,  u\  and  cr,  and  angle  opposite  u1  =  a1-  then 


sin  u  =  sin  u1  cos  <r  -\-  cos  u1  sin  cr  cos  or1, 
placing  the  serial  value  for  cos  <r  and  sin  <r, 

sin  u  =  sin  «'f  i .  .  .j  +  (<r  .  .  .)  cos  u1  cos  a1, 


omitting  all  powers  of  <r  above  the  second. 
Squaring  this  equation, 

sin"  u  —  sin*  ul(i  —  a3}  -j-  a*  cos"  w1  cos2  a1 

-{-  2<r  sin  u1  cos  «'  cos  a\ 

For  sin*  u,  write  I  —  cos8  u,  then  transpose,  change  signs,  mul 


234  GEODETIC  OPERATIONS. 

tiply  by  e\  subtract  from  I,  and  extract  the  square  root;  this 
gives 


r  I  —  e*  cos2  u  =  i  --  cos2  u1  -{-  e'er  sin  «'  cos  ul  cos  or1 


-j  --  o-2(cos2  «'  cos2  or1  —  sin2  w1).     (6) 


Placing  this  in  (5),  and  integrating  with  respect  to  dv,  we 
find 


s          t         ?  A  ,  ^V   . 

—  =  <rl  I  --  cos  u  I  -|  —  —  sin  «  cos  « 

s2  «l  cos"  or1  —  sin"  w1).     (7) 


cos  a 


This  can  be  written,  including  e\ 

s  (  f  ,  A 
—  =  <rl  i  --  cos  u  ) 
a  \  2  I 

r        f*  e*  "I 

I  -j  —  <r  sin  u1  cos  u1  cos  a1  -j-  ^-cr^cos4  u1  cosa  or1  —  sin"  «')    . 


If  we  place  a1  =  o,  we  have  <r  =  «  —  «',  then  the  last  equa- 
tion becomes,  after  placing  5  for  s, 


S  e1 

-  =(u  —  «')(i  —  -cos'w1) 


FIGURE   OF   THE  EARTH.  235 

From  sin  u  =  sin  «'( i j  -f-  cr  cos  «l  cos  #', 

\ve  get  by  transposition 

crs 
sin  u  —  sin  w1  =  cr  cos  «'  cos  a1 sin  #'.  (9) 

But  we  had 

#  —  w1  —  ;r,      or     u  =  ul-\-x;      hence     sin  u=  sin  (u1 -\-  ^r). 

Developing  this  by  Taylor's  formula, 

sin  u  =  sin  u1  -(-  ;r  cos  wl sin  w1 ;  (10) 


or        sin  u  —  sin  u1  =  x  cos  u1 sin  u1 

a* 
=  o-  cos  «'  cos  a1  —  —  sin  w1.      (l  i) 


Solving  this  equation  by  approximation, 

x  cos  u1  =  <T  cos  u1  cos  a1,          or        ^r  =  cr  cos  a1. 
Substituting  this  in  (il), 


.  .      ,     •         i  a  •  '          i 

x  cos  u1  —  -  cos"  a  sin  u  =  (r  cos  «  cos  a sin  «  ; 


236  GEODETIC  OPERATIONS. 

dividing  by  cos  u\ 


<r2                                                     <r9 
x cos2  a1  .  tan  u1  =  a  cos  a1 tan  u1. 

2  2 


a*  o-1 

x  =  <r  cos  a1 tan  ul  +  —  cos2  or1  tan  a1 

=  <r  cos  «J tan  ul(l  —  cos*  or1) 

=  cr  cos  a1 tan  ul  sin'  a1  =  u  —  ul;  (12) 


substituting  this  in  (8), 

-  =  (u  —  ul}( i  —  -  cos2 u1}}  i  -f-  -°" sin «'  cos  u1  cos  a1 

(I  \  2  /I 2 

-j (—  sin"  u1  -}-  2  cos2  u1  cos2  a1  -f-  sin2  «'  cos'  a1)  I.       (13) 

Dividing  this  equation  by  (7),  we  have 

5      u—  uT        So*  ~l 

-  = i (—3  sin2  u1  -f-  2  sin2  «'  -f-  sin2  u1  cos2  a1  I 

=  —^r-[_l  ~  £~^(~  sin!>  u*  +  sin'  wl(x  ~~  sin*  al)J 


FIGURE  OF  THE  EARTH.  237 

If  we  had  taken  ul  as  the  unknown  side  in  our  spherical  tri- 
angle, giving 


sin  ul  =  sin  u  cos  <r  -\-  cos  u  sin  <r  cos  a, 

we  would   have  found,  by  pursuing  a   course  similar  to  the 
above 

5       «_„'/        fa*   .  ,    \ 

-  =  — —\i-  —  sm  usm  a], 


from  which  we  could  obtain 

sin9  u1  sin"  a1  =  sina  «  sin"  or,        or        sin  w1  sin  a1=  sin  #  sin  or  ; 

hence  we  can  involve  both  the  direct  and  reverse  azimuth  as 
well  as  the  terminal  latitudes  by  writing  for  sin"  u1  sin2  a1,  sin 
u1  sin  a1  sin  u  sin  a,  so  that  (14)  will  become 


. 

I  I  ---  sm  u  sin  a  sin  u  sin  <*).         (15) 
12 


Resuming  the  former  notation,  we  will  put  a  =  Z,  and  or1  = 

Z1  —  1  80°;  also,  remembering  that  I  —  If  =  --  .  „  ,  ,  „.  --  ^r, 

sin3  ^(Z1  —  Z}' 

we  can  for  —  smZ  sin  Z1  write  sin'-K-Z"1  —  ^)  (i  —  #*),  and 
substitute  in  (15)  the  value  of  u  —  u1  in  (A),  which  gives 

—  =  h  (  i  ---  sin  w1  sin  u  sin  Z"1  sin  Z} 

S  \  12 

[i  -  ~  (i  -  70  +  ^  (i  -  ^s)(2  -  3^')}    (16) 


238  GEODETIC  OPERATIONS. 

This  still  involves  s  and  cr,  so  there  is  needed  a  relation  be- 
tween them.  To  attain  this  we  take  equation  (7),  using  only 
two  terms, 


€  \ 

=  <Tl  I  —  —  COS  U  COS  «'). 

Squaring  this,  and  omitting  terms  in  *?4, 

—  =  o"a(i  —  ^s  cos  «  cos  «l), 
j» 

Or  <7*  =  -YT 5 r>  (17) 

ar  (i  —  r  cos  «  cos  « ) 
Writing  /"  =  i  +  e*  cos  u  cos  w1, 

(16)  becomes 

5  =  sh\  (l -—^  sin  «'  sin  u  sin  Z1  sin  Z) 

^V"  /  sin  Z  sin  Z1  \         jV       sin  Z  sin  Z1   . 

/I  _i_  — ±__l 1 £ (2 ^A  ^ 

v     '    i2a*\sinl  i(Zl—  Z)/       240(1*  sin*  %(Zl  —  Zy        J    yj' 

sin|(Z+  Z1) 
in  which  h  =  -. — rr^i ^r- 


This  is  substantially  the  same  formula  as  given  by  Bessel  in 
Astronomische  Nachrichten,  No.  331,  pp.  309-10,  except  Z1  is 
within  the  polar  triangle,  which  gives 


-  Z1)' 


FIGURE  OF   THE  EARTH.  239 

sin  Z  sin  Z* 


and 


or  approximately 

Then,  writing 

/"  =  i  +  *"  cos  u  cos  u\  and  ^  =  i  -|-  e*  cos  (u  -f-  «'), 

Bessel's  formula  becomes 

sinZsin  Z' 

_  ^—-^ 

sin  ZsinZ1 


In  both  of  these  formulae  it  is  to  be  remembered  that 
tan  #  =  4/1  —  ^2  tan  Z,  and  tan  ul  =  Vi  —  e*  tan  U. 

Also,  if  the  line  deviates  but  little  from  the  meridian,  the 
first  term  will  be  sufficient. 

When  a  long  arc  has  been  measured,  it  has  been  found  best 
to  divide  it  into  several  sections,  from  each  of  which  data  can 
be  obtained  for  finding  the  axes  of  the  earth,  and  the  ellipticity. 
When  these  arcs  are  small,  the  method  given  on  page  228  will 
give  fair  results.  But  Clarke's  solution  is  perhaps  the  best  ;  it 
is,  in  the  main,  as  follows: 

Let  R,  x,  and  y  be  the  radius  of  curvature  of  an  ellipse,  and 

co-ordinates  of  the  point  whose  latitude  is  L,  then  -,  -f-  ^  =  I  ; 
but  we  have  shown  that 

x  =  a  cos  u,        tan  u  =  Vi  —  e*  tan  L, 


240  GEODETIC  OPERATIONS. 

from  which 

x  —  a  cos  L(i  —  e*  sin3  L)  -  *, 

also 

j/  =  £sin  u  =  #sin  L(\  —  e*)(i  —  e1  sin5  L}  -  *  ; 

tf=*(i-^)(i-*'sin'Z)-». 

Expanding,  and  neglecting  e', 


2  sin4  L  +  J/^4  sin4  Z). 
Substituting  I  —  cos2  L  for  sinZ3, 


Writing  A=a(i-&-  -&e<),  B  =  -  a(&+  &e<),  C  = 

we  have  R  —  A  +  2B  cos  2L  +  2^  cos  4^,  (B) 

This  is  an  ellipse  if  ^B1  =  6AC. 

Now,  if  5"  be  the  length  of  an  arc  of  an  elliptic  meridian,  it 
was  shown  on  page  231  that 

,c  .     r  .        ,  ,c       ^sin  udu 

aS  sin  L  =  a  sin  udu,  as  =  -  :  —  j-~. 

sin  L, 


FIGURE   OF   THE  EARTH.  241 

From  a  tan  u  —  b  tan  L,  we  found 


du  l/i  - 


therefore 


dL  ~  i  -  f  sin'  D 
ds  a  sin  u  Vi 


dL       sin  L(\  —  e*  sin'  L)' 
But  from  the  preceding  relation 


sin  L  V  i  — 
Sin  U  =  4 


Substituting  this,  we  have 


3Z 


by  integration, 

5  =  ^Z  +  ^sin2Z  •   \C  sin  4^+  a  constant. 


If  L  be  the  mean  latitude  of  an  arc  whose  amplitude  is  A, 
and  the  above  expression  be  integrated  between  the  limits 
L  —  ^A.  and  L  -(-  £A.,  we  will  obtain 

5"  =  A\.  -\-  2B  cos  2L  sin  A.  -)-  6*  cos  4^  sin  2/1.        (19) 

In  this  ^4,  B,  and  Care  the  only  unknown  quantities,  so  that 
if  S,  L,  and  A  be  free  from  errors,  three  equations  would  be  suf- 
ficient for  determining  A,  B,  or  C,  and,  consequently,  a,  e,  and 
16 


242  GEODETIC  OPERATIONS, 

b.  But  every  arc  is  affected  with  an  error,  in  length  as  well  as 
middle  and  terminal  latitudes,  so  that  from  a  number  of  dis- 
cordant results  we  must  find  the  most  probable  values  for  A, 
B,  and  C  by  the  principles  of  least  squares. 

Suppose  the  terminal  latitudes  have  a  small  error  in  each  of 
x,  and  xf,  so  that  the  amplitude  would  be  A  -j-  x*  —  x^  and 
the  latitudes  L  —  £A  +*„  and  L  —  £A  -f-  *,x. 

Placing  these  corrected  values  in  (19), 


s  2L  sin  (A  +  */  -  *,) 

C  cos  4^  sin  2(A  -f-  .ar,1  —  jr,).     (20) 


In  expanding  this  we  treat  x?  —  ;r,  as  a  single  term,  and 
being  small,  cos  (x*  —  x^  =  I,  and  sin  (x*  —  x^)  =  x*  —  x^  so 

sin  (\-\-x  ^—  x^)  =  sin  A  -)-  cos  A(jtr/  —  x^}  ; 

sin  2(A.+;r1I-*1)=  2  sin  [A  +  (jr/  -  ^)]  cos  [A  -f  (**  -  ^)] 

=  2[sin  X-l-cosA^,1—  ^^[cos  A—  sin  A(^/—  ^)] 
=  sin  2  A  -}-  2(jr,'  -r-  x^  cos  2A  ; 

substituting  these  expressions  in  (20), 

S=A(l+  */  -  ^)  +  2,5  cos  2Z[sin  A  +  cos  Afo1  -  JT,)] 

+  Ccos  4^[sin  2A  -}-  2(^,1  —  #,)  cos  2A]. 

Solving  for  ^r,1  —  ^r, 


cos  2A)  =  5  —  ^A 
—  2Bcos2L  sin  A  —  (Tcos^Zsin  2A.     (21) 


If  we  write  A-\-2B  cos  A  cos  2Z,  2(7  cos  ^L  cos  2A  =  —  ,  (2  1) 
will  reduce  to 


FIGURE   OF   THE  EARTH.  243 


IS          _\            2BH    . 
',    —  x^  —  Irs-  —  A  J,u —  sin  A  cos  2L 


—f-  sin  2!  cos  4Z.      (22) 


Expressing  ^r,1,  .*•„  and  A  in  seconds — we  approximate  the 
length  of  a  second  of  latitude  by  assuming  the  average  radius 
of  curvature  to  be  20855500  ft. — we  must  write 

v=  2085 5  500  sin  i". 

Then  we  assume  three  auxiliary  quantities,  u,  v,  and  Z,  and 
place 


A       20855500\ 

_  —  -  —  _i_      v 

A  nr^~    ~\ 


A          2OO    '     IOOOO 

C          Z 


A       loooo" 
Substituting  these,  (22)  becomes 


IOOOOU 


_  1  _U  sin  ^  cos  2^ 

2oo  sin  i" 


sin  A  cos  2Zz/       sin  2A  cos 
"~  ~ioooo  sin  i"  "^      loooo  sin  i"  '     ^  ^ 


Again  we  assume 

IS  sin  A  cos  2L 


244  GEODETIC  OPERATIONS. 

sin  \  cos  2L^i  sin  2\  cos 


10000  sin  i"'  '    IOGOO  sin  i"  ' 

fj.  =  I  -f-  -g-^-jj-  cos  A.  cos  zL. 

Then  (23)  can  be  written 

x*  —  x^  =  m  -\-  au  -\-  bv  -f-  ^.Z, 
or  #,'  =  ;r1  +  a«4-<«<  +  ^  +  c^-  (24) 


For  each  arc  or  partial  arc  we  will  have  an  equation  like  (24), 
which  is  to  be  solved  by  the  principles  of  least  squares,  by 
making  the  sum  of  the  squares  of  the  errors  a  minimum  ;  then 
equating  the  differential  coefficients  of  the  symbolic  errors 
with  respect  to  u,  v,  z,  x?,  etc.,  to  zero,  there  will  be  as  many 
equations  as  there  are  unknown  quantities  to  be  solved  by  al- 
gebraic methods.  Knowing  u,  v,  and  z,  we  find  A,  B,  and  C, 
which  substituted  in  (B)  give  R. 

To  determine  the  axes  and  ellipticity,  we  take  the  equations 
on  page  240  and  find  that  the  coefficient  of  cos  L  =  (A  —  .Z?), 
of  cos  3-£  =  %(B  —  C),  and  of  cos  t>L  =  \C\  also,  of  sin  L  — 
A  -\-B,  of  sin  3^  =  #JB  +Q,  and  of  sin  *>L  —  \C.  By  making 
these  substitutions,  we  have 


x-  (A  -  B)  cos  L  +  #B  —  <7)cos3£+Kcos5Z;    (25) 

)  sin  3^  +  \C  sin  5^.     (26) 


But  on  page  231, 

x  =  —fa  sin  L  dL,        and        y  =fbcos  L  dL. 


FIGURE   OF   THE  EARTH.  245 

So  (25)  and  (26)  are  the  values  of  these  integrals,  which  if  in- 
tegrated between  the  limits  L  =  o  and  L  =  90°,  will  give 
the  semi-axes, 


(27) 
(28) 


V  SJBf     , 

=i-      =    -          i  +         ,  approximately. 


If  these  values  be  substituted  in  (25)  and  (26),  we  would  have 


(30) 


(29)  and  (30)  are  the  values  of  the  co-ordinates  of  a  point  in 
an  elliptic  curve  whose  axes  are  a  and  b,  while  (25)  and  (26) 
are  the  co-ordinates  of  a  point  in  the  actual  curve.  The  dif- 
ference between  the  two  will  be  the  deviation  of  the  actual 
from  the  elliptic  curve  at  any  point. 


I  I 

x  —  xl  =  { C  —  ~ }  ( —  cos  L  —  -  cos  3^  -f  -  cos 


246 


GEODETIC  OPERATIONS. 


Si-     (32) 


Suppose  P  be  the  point  on  the 
elliptic  curve  in  latitude  Z,  and  Q 
the  point  on  the  actual  curve  in  the 
same  latitude.  P  and  Q  will  coin- 

t£? 
cide  when  C  —  g-^-  =  o,  for  this  will 

reduce  (31)  and  (32)  to  x  —  x'  =  o, 
—  y1  —  o,  and  will  differ  from  one 

$1? 

another  as  C  —  ^-v-  changes    from 

a  zero  value. 

If  we  take  PS  an  infinitesimal 
distance  on  the  elliptic  curve,  and  QS  a  corresponding  length 
along  the  normal, , we  wilt  have 


FIG.  31. 


=  y-y 


=  x-x\ 


QS=  QU+SU=  QU+PV 

—  (x  —  x1)  cos  L  -f-  (y  —  j/1)  sin  L, 


or         dR  —  (x  —  x1)  cos  L  -f  L(y  —  / )  sin  L. 


(33) 


PS=  VU=  TU-  TV 

—  —  (x  —  x1}  sin  L  -|-  (y  —  j1)  cos  L, 


$  =  —  (x  —  x1}  sin  L  -\-  (y  — j/1)  cos 


(34) 


FIGURE  OF   THE  EARTH.  247 

Substituting  in  these  equations  the  values  of  (x  —  xl~]  and 
—  y)  from  (31)  and  (32),  we  find 


$& 

Clarke's  values  of  a  and  b  of  1866  would  give 


showing  but  a  slight  deviation  of  a  meridian  section  from  an 
ellipse. 

The  Anglo-French  arc  places  the  actual  curve  3.6  feet  under 
the  ellipse  in  latitude  58°,  and  18.9  feet  above  in  latitude  44°  ; 
while  the  Indian  arc  places  it  19.6  feet  under,  in  latitude  14°, 
and  9.3  feet  above,  in  latitude  26°. 

The  amplitude  of  an  arc  depending  upon  the  latitude  deter- 
minations of  its  extremities  is  subject  to  an  error  from  local 
deflection.  In  some  cases,  at  least  a  portion  of  these  errors 
can  be  corrected  by  computing  the  effects  of  attraction  upon  a 
physical  hypothesis;  but  in  the  main  they  are  best  treated 
as  accidental,  and  the  figure  of  the  earth  determined  by  the 
principle  of  least  squares,  in  which  the  sum  of  the  squares 
of  all  errors  shall  be  a  minimum. 

This  was  suggested  by  Walbeck  in  1819,  continued  by 
Schmidt  in  1829,  and  perfected  by  Bessel  in  1837. 

Laplace  in  1822,  published  the  second  volume  of  Mtcanique 
Celeste,  in  which  he  discussed  the  figure  of  the  earth,  using 
seven  arcs:  the  Peruvian,  Lacaille's  Cape  of  Good  Hope  arc, 
Mason  and  Dixon's,  Boscovich's  Italian,  Delambre  and  Me- 
chain's,  Maupertuis'  Lapland  arc,  and  Liesganig's  Austrian  arc. 


248  GEODETIC  OPERATIONS. 

The  second  is  unreliable,  from  an  erroneously  assumed  cor- 
rection for  local  attraction  which  shortened  the  arc  by  9"  too 
much.  The  third  was  a  measured  arc,  and  not  comparable 
with  a  trigonometric  one.  And  no  confidence  is  now  placed 
in  either  the  fourth  or  the  last. 

Bowditch,  in  his  translation  of  the  above-named  work,  con- 
siders only  the  Peru  and  France  arcs,  and  adds  those  of  Eng- 
land and  India  as  completed  in  1832.  His  conclusion  is: 

"  It  appears  that  this  strictly  elliptical  form  of  the  meridian 
is  more  conformable  to  these  observations  than  the  irregular 
figure  obtained  by  Mr.  Airy's  calculation." 

Sir  George  Airy  published  in  the  Encyclopedia  Metropoli- 
tana,  under  the  heading  "  Figure  of  the  Earth,"  in  1830,  a  dis- 
cussion of  fourteen  meridian  arcs  and  four  arcs  of  parallel. 
In  1841,  Bessel  gave  to  the  public  the  results  of  his  laborious 
investigation  of  ten  meridian  arcs,  having  a  total  amplitude  of 
5o°-5,  and  embracing  thirty-eight  latitude  stations.  The  re- 
sult gave  an  elliptic  meridian,  and  the  elements  then  published 
are  still  known  as  those  of  Bessel's  spheroid. 

In  1858,  in  the  "Account  of  the  Principal  Triangulation  of 
Great  Britain  and  Ireland,"  Captain  Clarke  gives  a  most  elabo- 
rate discussion  of  eight  arcs,  having  a  total  amplitude  of  78° 
36',  and  embracing  sixty-six  latitude  stations. 

Again,  in  1880,  he  revised  his  previous  computations,  using 
corrected  positions  from  which  slightly  different  results  were 
obtained. 

Mr.  Schott  discussed  the  combination  of  three  American 
arcs  of  meridian  for  determining  the  figure  of  the  earth  con- 
sidered as  a  spheroid.  He  used  the  Pamlico-Chesapeake,  Nan- 
tucket,  and  Peruvian,  having  a  total  amplitude  of  n°oi'  12", 
and  embracing  twenty-three  latitudes.  The  conclusion  de- 
duced by  Mr.  Schott  is  :  "  The  result  from  the  combination  of 
the  three  American  arcs  is  the  preference  it  gives  to  Clarke's 
spheroid  over  that  of  Bessel." 


FIGURE   OF   THE  EARTH. 


249 


TABLE  GIVING  THE    ELLIPTICITY    AND    LENGTH    OF    A    QUAD- 
RANT ON  THE  SPHEROIDAL  HYPOTHESIS. 


Date. 

Authority. 

Ellipticity. 

Quadrant  in  Metres. 

1819 

Walbeck  

•  go2  8 

68 
O   OOO   2OO 

1810 

Schmidt 

18^0 

Airy   

1841 

Bessel  

10  ooo  976 

1856 

Clarke  

10  ooo  050 

1863 

Pratt  

10  ooi  515 

1866 

Clarke  

1868 

Fischer  

•  288  5 

1872 

•  °Sa 

a 

1877 

10   OOO    218 

1878 

•  286  5 

10   002    232 

1880 

Clarke     

10   000   081 

9 

Data  for  the  Figure  of 
the  Earth. 

Bessel,  1841. 

Clarke.  1866. 

Coast  Survey,      Clark      ,88o 
1877. 

Equatorial  radius,  a.  . 
Polar  semi-axis,  b.  .  .  . 
a  —  b 
Compression,    —  —  .. 

Mean  length  of  a  deg. 

6377397-2M 
6356079 

i  :  299.15 
in  I2O.6M 

6378  206.  4M 
6356583-8 

i  :  294.98 
in  132.1 

6378054.3X1 
6357175 

I  :  305-4S 
i"  135-9 

6  378  248.  5M 
6  356  514.  ?M 

i  :  293.5 
in  131.8 

The  value  of  the  ellipticity  as  deduced  by  pendulum-obser- 
vations in  accordance  with  Clairaut's  theorem  is  I  :  292.2,  be- 
ing almost  the  same  as  that  obtained  from  geodetic  measure- 
ments. 

Clarke's  length  of  the  quadrant  would  give  for  the  metre 
39.377786  inches,  whereas  the  legal  length  is  39.370432  inches, 
or  .0073  inch  too  short. 


LITERATURE  OF  THE  FIGURE  OF  THE  EARTH. 

Pratt,  A  Treatise  on  Attractions,  Laplace's  Functions,  and 
the  Figure  of  the  Earth.  London,  1861. 

Roberts,  Figure  of  the  Earth.  Van  Nostrand 's  Engineering 
Magazine,  vol.  xxxii.,.pp.  228-242. 


2$O  GEODETIC  OPERATIONS. 

Merriman,  Figure  of  the  Earth.     New  York,  1881. 

U.  S.  Coast  Survey  Report  for  1868,  pp.  147-153. 

U.  S.  Coast  and  Geodetic  Survey  Report  for  18/7,  pp.  84-95. 

Clarke,  Geodesy,  pp.  302-322.     London,  1880. 

Laplace,  Mecanique  Celeste.  Bowditch's  Translation,  vol. 
ii.,  pp.  358-485.  Boston,  1830. 

Ordnance  Survey,  Account  of  Principal  Triangulation,  pp. 
733-782.  London,  1858. 

Bruns,  Die  Figur  der  Erde.     Berlin,  1878. 

Baeyer,  Grosse  und  Figur  der  Erde.     Berlin,  1861. 

Jordan,  Handbuch  der  Vermessungskunde,  vol.  ii.,  pp.  377- 
463.  Stuttgart,  1878. 


FORMUL/E   AND    FACTORS. 


FORMULA  AND  FACTORS.  253 

TRIGONOMETRIC   EXPRESSIONS. 
sin2  a  -f-  cos3  a  =  i  ; 


sin  a  =  i/i  —  cos2  a 
_  cos  # 
~  cot  a 


Vi  +  cot*  a 
=  cos  a  tan  a 
=  2  sin  \a  cos  | 

I 
~~  cosec  a 


sec  a 


sin  a 
tan  a  —  — 

cos  a 


cot  a 


254  GEODETIC  OPERATIONS. 

sin  a 


r  i  —  sin  a 
i  —  cos  2a 


sin  2a 


i  —  cos  2a 


cot  *  = 


i  -|-  cos  2a 
I 


cosec  a  = 


versin  a  —  i  —  cos  0  —  2  sin9  \a 
chord  <z  =  2  sin  \a. 
sin  («  ±  V)  —  si"  «  cos  b  ±  cos  «  sin  b  ; 

cos  («  ±  #)  =  cos  a  cos  *  ^  sin  *  sin  ^' 

tan  ^  ±  tan  b 
tan  («  ±  0)  =  Y^te^'a  tan  b' 


cot  #  cot 
cot  («  ±  *)  =     cot  ^  ± 


sin  2^  =  2  sin  #  cos  d!. 

cos  2a  —  cosa  #  —  sin2  a 

=  2  cos*  a  •—  I  =  I  —  2  sin" 


FORMULA  AND  FACTORS.  255 

2  cos4  %a  =  i  -}-  cos  0. 


2  sin2  ^  =  i  —  cos  a. 
i  —  cos  a 


tan2    *  = 


I  -f-  COS  tf 


sm  a  ±  sm  b  =  2  sin  $(a  ±  b)  cos  £( 
cos  #  -j-  cos  b  —  2  cos  i(tf  -j-  £)  cos%(a  —  b]. 
cos  a  —  cos  #  =  2  sin  £(#  -f-  b]  sin  £(£  —  a). 
sin2  a  —  sin2  b  =  sin  (a  -\-  b)  sin  (a  —  b). 
cos2  a  —  sin2  b  =  cos  (a  -J-  £)  cos  (#  —  #). 

sin  2x  =  2  sin  JT  cos  jr. 

sin  3jr  =  3  sin  jr  —  4  sin3  jr. 

sin  4jr  =  (4  sin  JT  —  8  sin3jr)  Vi  —  sin2  jr. 

cos  2x  =  cos2  JT  —  sin2  x  =  2  cos2  JT  —  i. 

cos  3jr  =  4  cos3  JT  —  3  cos  jr. 

cos  4jr  =  8  cos4  JT  —  8  cos2  x  -{-  i. 

2  tan  jr 
tan  2x  = -r— . 

i  —  tan  JT 

3  tan  JT  —  tan3  jr 

tan  3jr  = 5 . 

i  —  3  tan   jr 

4  tan  jr  —  4  tan3  x 
tan  4x  = -£-  .  --—T-. 


256  GEODETIC  OPERATIONS. 

TRIGONOMETRIC   SERIES. 


x' 


X"  2X"  \TX 

f+-+  —  +  7- 

I          X  X*  2X* 


i    . 

cosec,  =  _  +  _+_,_+__ 


,    sin8  x        3  sin5  ^,3.5  sin' 

arc"  =  sm"  +  ^T  +  ^:T  +  ^^6 

=  tan  x  —  \  tan3  x  -\-  ^  tan5  ^  .  .  .  . 
For  very  small  angles  Maskelyne's  series  is  best. 


«  /  -  ,  (        **       x'\ 

=  x  V  cos  ^r  -]-...=  ^rl  i  —  ->-  ---  J. 


BINOMIAL,    EXPONENTIAL,    AND   LOGARITHMIC   SERIES. 


^"  ^ 

'  aY72  +  log'  *273 


FORMULAE  AND  FACTORS. 
log  (X  +  I)  : 

M  =  nodulus  =  0.4342945. 
log  M  =  9-6377843. 

CONVERSION  OF  METRES  TO  FEET. 

Metres  X  3.280869        =  feet,  or  to  log  of  metres  add  0.5159889 
X  1.093623        —yards,  "  0.038867 

X  0.000621377  =  mile,  "  "  6.7933550-10. 

i  toise  =  76.734402  inches  =  864  lines. 

i  Prussian  foot  =  139.13  lines. 
i  klafter  =  840.76134  lines. 

The  toise  is  that  of  Peru,  which  is  a  standard  at  13°  R. 


CONVERSION  TABLES. 

METRES   INTO   YARDS. 

i  metre  =  1.093623  yards. 


Metres. 

Yards. 

Metres. 

Yards. 

Metres. 

Yards. 

IOO   OOO 

109  362.3 

3  000 

3  280.87 

60 

65-617 

90  ooo 

98  426.1 

2   OOO 

2    187.25 

50 

54.681 

80  ooo 

87  489.8 

I    OOO 

I   093.62 

40 

43  •  745 

70  ooo 

76  553.6 

900 

984.26 

30 

32.809 

60  ooo 

65  617.4 

800 

874.90 

2O 

21.872 

50  ooo 

54  681.2 

700 

765.54 

10 

10.936 

40  ooo 

43  744-9 

600 

656.17 

9 

9-843 

30  ooo 

32  808.7 

500 

546.81 

8 

8.749 

20   000 

21    872.5 

400 

437-45 

7 

7-655 

10   000 

10  036.2 

300 

328.09 

6 

6.562 

9  ooo 

9  842.61 

2OO 

218.72 

5 

5.468 

8  ooo 

8  748.98 

IOO 

109.36 

4 

4-374 

7  ooo 

7  655.36 

90 

98.426 

3 

3.281 

6  ooo 

6  561.74 

80 

87.490 

2 

2.187 

5  ooo 

5  468.12 

70 

76.554 

I 

1.094 

4  ooo 

4  374-49 

258 


GEODETIC  OPERATIONS. 


CONVERSION  TABLES — Continued. 

YARDS    INTO    METRES. 

i  yard  =  0.914392  metre. 


Yards. 

Metres. 

Yards. 

Metres. 

Yards. 

Metres. 

IOO   000 

91  439.2 

3  000 

2   743-18 

60 

54.864 

90  ooo 

82  295.3 

2   OOO 

I   828.78 

50 

45-720 

80  ooo 

73  I5I-3 

I   OOO 

9I4-39 

40 

36.576 

70  ooo 

64  007.4 

900 

822.95 

30 

27.432 

60  ooo 

54  863.5 

800 

73L5I 

2O 

18.288 

50  ooo 

45  7I9-6 

700 

640  .  07 

IO 

9.144 

40  ooo 

36  575-7 

600 

548.64 

9 

8.230 

30  ooo 

27  431-8 

500 

457.20 

8 

7.315 

20  ooo 

18  287.8 

400 

365.76 

7 

6.401 

10   000 

9  143-9 

300 

274.32 

6 

5-486 

9  ooo 

8  229.53 

2OO 

182.88 

5 

4-572 

8  ooo 

7  3I5-I3 

IOO 

91.44 

4 

3.658 

7  ooo 

6  400.74 

90 

82.295 

3 

2-743 

6  ooo 

5  486.35 

80 

73.I5I 

2 

1.829 

5  ooo 

4  57I-96 

70 

64.007 

I 

0.914 

4  ooo 

3  657.57 

METRES    INTO    STATUTE    AND    NAUTICAL   MILES. 

I  metre  =  0.00062138  statute  mile. 
i  metre  =  o  00053959  nautical  mile. 


Metres. 

Statute  Miles. 

Nautical  Miles. 

Metres. 

Statute  Miles. 

Nautical  Miles. 

IOO  000  |          62.138 

53-959 

900 

0-559 

0.486 

90  ooo  ,          55.924 

48.563 

800 

0.497 

0.432 

80  ooo  |         49-710 

43.167 

700 

0-435 

0.378 

70  ooo  .        43-496 

37-772 

600 

0-373 

0.324 

60  ooo  !         37.283 

32.376 

500 

0.311 

0.270 

50  ooo 

31-069 

26.980 

400 

0.249 

0.216 

40  ooo 

24.855 

21.584 

300 

0.186 

0.162 

30  ooo 

18.641 

16.188 

200 

0.124 

0.108 

20   OOO 

12.428 

10.792 

IOO 

0.062 

0.054 

IO  OOO 

6.214 

5-396 

90 

0.056 

0.049 

9  ooo 

5-592 

4-856 

80 

0.050 

0.043 

8  ooo 

4-971 

4-3I7 

70 

0.043 

0.038 

7  ooo 

4-350 

3-777 

60 

0.037 

0.032 

6  ooo 

3.728 

3-238 

50 

0.031 

0.027 

5  ooo 

3.107 

2.698 

40 

0.025 

0.022 

4  ooo 

2.486 

2.158 

30 

0.019 

0.016 

3  ooo 

1.864 

1.619 

2O 

O.OI2 

O.OII 

2    000 

1-243 

1.079 

IO 

O.OO6 

0.005 

I    OOO 

O.62I 

o  540 

FORMULA  AND  FACTORS. 


259 


CONVERSION  TABLES — Continued. 

STATUTE  AND   NAUTICAL  MILES   INTO   METRES. 

i  statute  mile    =  1609.330  metres, 
i  nautical  mile  =  1853.248  metres. 


Miles. 

Metres  in 
Statute  Miles. 

Metres  in 

Nautical  .Miles. 

Miles. 

Metres  in               Metres  in 
Statute  Miles.      Nautical  Miles. 

100 

160  933-0 

185   324-8 

•9 

I  448.40             I  667.92 

9° 

144  839.7 

166  792.3 

.8 

I   287.46             I  482.60 

80 

128  746.4 

I48   259.8 

•  7 

I   126.53             I   297.27 

70 

112    653.  I 

129   727.4 

.6 

965.60             I   111.95 

60 

96  559-8 

III    194.9 

-5 

804.67                 926.62 

50 

80  466.5 

92   662.4 

-4 

643-73 

741   30 

40 

64  373-2 

74  129-9 

-3 

482.80 

555-97 

30 

48  279.9 

55  597-4 

.2 

321.87 

370.65 

20 

32  186.6 

37  065.0 

.1 

160.93 

185-32 

10 

16  093.3            18  532.5 

.09 

144.84 

166.79 

9 

14  483-97          l6  679.23 

.08 

128.75 

148.26 

8 

12  874.64          14  825.98 

.07 

112.65 

129.73 

7 

ii  265.31          12  972.74 

.06 

96.56 

111.19 

6 

g  655.98          ii   119.49 

•05 

80.47 

92.66 

5 

8  046.65           9  266.24 

.04 

64-37 

74-13 

4 

6  437-32           7  412.99 

•03 

48.28 

55.60 

3 

4  827.99  ;         5  559-74 

.02 

32.19 

37.06 

2 

3  218.66 

3  706.50 

.OI 

16.09 

18.53 

I 

i  609.33 

i  853.25 

Major   semi  axis  =  a,  minor   semi-axis  =  b,  ellipticity  =  e 
a-b 


Bessel,     a  —  6377397  .  i$M,         log  =  7.8046434637 
b  =  6356078  .  g6M,        log  =  6.8031892839 


Clarke,  a  —  6378206  .  4^f, 
b  =  6356583  .  SM, 
e  =  m- 


log  =  6.8046985352 
log  =  6.8032237974 


260 


GEODETIC  OPERATIONS. 


CONSTANTS    AND    THEIR    LOGARITHMS. 


Number. 

Log. 

Ratio  of  circum.  to  diameter, 

Tt  3.1415926 

0.4971499 

27T  6.2831853 

0.7981799 

TT"  9,-8696o44 

0.9942997 

Vn  1.7724538 

0.2485749 

Number  of  degrees  in  circum. 

360 

2.5563025 

Number  of  minutes  in  circum 

.,              21600 

4-3344538 

Number  of  seconds  in  circum, 

.,          1296000 

6.1  126050 

Degrees  in  arc  equal  radius,          57°-295779 
Minutes  in  arc  equal  radius,      3437  -7467 
Seconds  in  arc  equal  radius,  206264  .806 


1.7581226 


Length  of  arc  of  I  degree, 
Length  of  arc  of  I  minute, 
Length  of  arc  of  i  second, 

Naperian  base, 

sin  i" 
*  sin  i" 


•0174533 
.0002909 
.00000485 


5.3144251 

8.2418774—  10 
6.4637261  —  10 
4.6855749  -  10 


2.7182818       0.4342945 

4.6855749 
4.3845449 


N  is  the  normal  produced  to  the  minor  axis.  R  is  the  radius 
of  curvature  in  the  meridian.  Radius  of  curvature  of  the 
parallel  is  equal  to  N  cos  /,. 

The  following  tables  are  based  upon  Clarke's  spheroid  of 
1866,  and  were  computed  in  1882.  Since  then  similar  tables 
have  been  published  by  the  Geodetic  Survey,  with  which  the 
appended  have  been  compared. 


FORMULAE  AND  FACTORS. 


26l 


Lat. 

V~(i  -  ,»sin«£)f 
Log  JV. 

«d  -  *") 

Log(i+^cos'Z,). 

-  (i-,*sin«£)l- 
Logtf. 

24°00' 

6.8049418 

6.8024790 

0.0024628 

10 

9450 

4884 

4566 

2O 

9481 

4981 

4500 

30 

9512 

5076 

4436 

40 

9545 

5174 

4371 

50 

9577 

5270 

4307 

25  oo 

9612 

5370 

4242 

IO 

9645 

5470 

4175 

20 

96/7 

5569 

4108 

30 

9711 

5667 

4044 

40 

9744 

5768 

3976 

50 

9777 

5869 

3908 

26  00 

9812 

5968 

3841 

10 

9846 

6070 

3774 

20 

9880 

6173 

3706 

30 

9915 

62/6 

3639 

40 

9948 

6379 

3569 

50 

9981 

6482 

3499 

27  oo 

6.8050017 

6585 

3432 

IO 

0051 

6688 

3363 

20 

0086 

6794 

3292 

30 

OI2O 

6899 

3221 

40 

0156 

7006 

3150 

50 

Oigl 

7111 

3080 

28  oo 

O227 

7216 

3011 

IO 

0263 

7322 

2941 

20 

0299 

7429 

2870 

30 

0334 

7537 

2/97 

40 

0371 

7644 

2727 

5° 

0407 

7752 

2655 

29  oo 

0444 

7862 

2582 

10 

0480 

797i 

2509 

20 

0517 

8081 

2436 

3° 

0555 

8187 

2368 

4° 

0591 

8296 

2295 

5° 

0628 

8408 

222O 

30  oo 

0664 

8524 

2I4O 

10 

O7OO 

8636 

2064 

20 

0738 

8747 

iggi 

3° 

0776 

8858 

igiS 

40 

0813 

8972 

I84I 

50 

0849 

9084 

1765 

31  oo 

089! 

9198 

1693 

10 

0928 

9310 

1618 

20 

0976 

9426 

1550 

30 

IOI4 

9540 

1474 

4° 

1054 

9654 

1400 

50 

1089 

9769 

1320 

262 


GEODETIC  OPERATIONS. 


Lat. 

jV  —       a 

?      "(i-**) 

Logd+^cos'Z,)- 

(i  -*»sin*  A)f 
Log  N. 

(i  -  «»  sin"  L)f 

Log;?. 

32°oo' 

6.8051128 

6.8029885 

0.0021243 

10 

1166 

6  .  8030002 

1164 

20 

1205 

0117 

1088 

30 

1244 

0232 

1012 

40 

1283 

0349 

0934 

50 

1322 

0466 

0856 

33  oo 

1351 

0583 

0768 

10 

1390 

0700 

0690 

20 

1429 

0818 

0611 

3° 

1469 

0937 

0532 

40 

1508 

1055 

0453 

50 

1548 

1174 

°374 

34  oo 

1587 

1293 

0294 

10 

1627 

1414 

0213 

20 

1667 

1532 

oi35 

30 

1707 

1652 

0055 

40 

1746 

1769 

0.0019977 

50 

1785 

1889 

9896 

35  oo 

1828 

2014 

9814 

10 

1868 

2134 

9734 

20 

1909 

2255 

9654 

30 

1949 

2376 

9573 

40 

2989 

2499 

9490 

50 

2029 

2619 

9410 

36  oo 

2070 

2743 

9327 

10 

2III 

2865 

9246 

20 

2152 

2987 

9165 

30 

2192 

3110 

9082 

40 

2233 

3234 

8999 

5° 

2274 

3354 

8920 

37  oo 

2316 

3480 

8836 

10 

2358 

3602 

8756 

20 

2398 

3727 

8671 

30 

2440 

3851 

8589 

40 

2482 

3975 

8507 

50 

2523 

4098 

8425 

38  oo 

2505 

4225 

S340 

IO 

2607 

4350 

8257 

20 

2648 

4475 

8i73 

30 

260X) 

4599 

8091 

40 

2732 

4726 

8006 

50 

2775 

4846 

7929 

39  oo 

2815 

4977 

7838 

10 

2857 

5102 

7755 

20 

2899 

5228 

7671 

30 

2941 

5355 

7586 

40 

2984 

5482 

7502 

50 

3025 

5608 

7417 

FORMULA  AND  FACTORS. 


263 


Lat. 

a 

«(!-*») 

U.C.+*-* 

Log  AT. 

(i  -  *»  sin*  L)f 
Log  R. 

4O°Oo' 

6.8053068 

6-8035734 

0.0017334 

IO 

3"i 

5859 

7252 

20 

3154 

5987 

7167 

30 

6115 

7080 

40 

3237 

6242 

6995 

50 

3280 

6367 

6913 

41  oo 

3321 

6497 

6824 

10 

3365 

6625 

6740 

20 

3407 

6752 

6655 

3° 

3450 

6880 

6570 

40 

3592 

7008 

6484 

50 

3535 

7130 

6405 

42  oo 

3577 

7263 

6294 

IO 

3620 

7392 

6228 

20 

3663 

7519 

6144 

30 

3706 

7649 

6057 

40 

3749 

7777 

5972 

50 

3792 

79°5 

5887 

43  oo 

3832 

8032 

5802 

IO 

3877 

8160 

5717 

20 

39*9 

8288 

5631 

3° 

3962 

8417 

5545 

4° 

4004 

8549 

5455 

50 

4047 

8680 

53^7 

44  oo 

4090 

8803 

5287 

10 

4134 

8930 

5204 

20 

4177 

9059 

5116 

30 

4219 

9188 

5031 

40 

4262 

9317 

4945 

50 

4306 

9445 

4861 

45  oo 

4347 

9575 

4772 

10 

439* 

9704 

4687 

20 

4434 

9834 

4600 

30 
40 

4477 

9961 
6.8040090 

429 

50 

4563 

0218 

4345 

46  oo 

4604 

0347 

4258 

10 

4648 

0476 

4172 

20 

4690 

0605 

4085 

3° 

4734 

0734 

4000 

40 

50 

4777 
4820 

0860 
0989 

3917 
3831 

47  oo 

4861 

1118 

3744 

IO 

4905 

1247 

3658 

20 

4948 

1376 

3572 

3° 

4991           1504 

3487 

4° 

5033 

1631 

3402 

50 

5076 

1759 

3317 

264 


GEODETIC  OPERATIONS. 


Lat. 

v      a 

R    «d-<') 

Log  (i  +  <r'  cos"  L). 

(i  -  *  sin*  L* 
LogJV. 

1n-  +  **L]K 

LogK. 

4S°oo' 

6.8055118 

6.8041887 

0.0013231 

10 

5l6o 

2016 

3144 

2O 

5202 

2144 

3058 

30 

5244 

2272 

2972 

40 

5289 

2400 

2889 

50 

5333 

2528 

2805 

49  oo 

5374 

2657 

2717 

10 

5417 

2784 

2633 

20 

5459 

2909 

2550 

3° 

5501 

3037 

2464 

40 

5545 

3163 

2382 

50 

5587 

3293 

2294 

50  oo 

5629 

3418 

2211 

10 

5672 

3544 

2128 

20 

57H 

3671 

2043 

30 

5756 

3798 

1958 

40 

5798 

3925 

1873 

50 

5841 

4048 

1790 

THE  A,  B,   C,  £>,  E   GEODETIC    FACTORS. 
From  latitude  24°  to  48°,  inclusive. 

A  = 


D  

c= 

£>  = 


2NR  arc  i" 

|^3  sin  L  cos  L 
(i  -^sTiTZ)!' 

I  4-  3  tan3  L 


Referred  to  Clarke's  spheroid  of  1866. 


FORMULA  AND  FACTORS. 


265 


Lat. 

Log  A. 

Log  B.        Log  C. 

Log/?. 

Log  E. 

24°00' 

8.5094834 

8.5119462 

1.05456 

2.2629 

5.8147 

05 

818 

415 

625 

40 

59 

IO 

802 

368 

794 

52 

72 

15 

786 

320 

962 

64 

85 

20 

769 

271 

1.06130 

75 

97 

25 

753 

223 

297 

86 

5.8210 

30 

738 

174 

464 

97 

23 

35 

720 

127 

631 

2.2708 

36 

40 

704 

078 

797 

19 

49 

45         688 

028 

962 

30 

62 

50        672 

8.5118979 

1.07128 

40 

74 

55         659 

030 

-93 

5i 

87 

25  oo        640 

882 

457 

62 

5-8300 

05         623 

833 

621 

72 

13 

10        607 

782 

785 

83 

26 

15 

591 

733 

948 

93 

39 

20 

573 

684 

i.  08111 

2.2804 

52 

25 

556 

634 

274 

15 

66 

30 

541 

585 

435 

25 

79 

35 

524 

535 

597 

35 

92 

40 

508 

484 

759 

45 

5-8405 

45 

491 

437 

920 

55 

18 

5° 

473 

383 

1.09080 

•65 

31 

55         456 

337 

241 

75 

45 

26  oo        440 

283 

400 

85 

58 

05         423 

232        560 

95 

7i 

10        406 

181 

719 

2.2905 

85 

15         ^88 

130 

878 

15 

98 

20 

372 

078 

i  .  10036 

24 

5-8512 

25 

354 

027 

194 

34 

25 

3° 

337 

8.5"7977 

352 

44 

39 

35 

320 

924 

509 

53 

52 

40 

3°3 

874 

666 

63 

66 

45 

287 

811 

854 

72 

79 

50 

55 

270 
252 

770 
718 

979 

i.  11135 

81 
91 

93 
5.8606 

27  oo 

235 

667 

290 

2.3000 

20 

05 

218 

616 

445 

09 

34 

10 

201 

564 

600 

18 

47 

15 

182 

5" 

755 

27 

6l 

2O 

1  66 

458 

909 

36 

75 

25 
30 
35 

148 
132 
H3 

405 
353 
310 

1.12063 
217 
370 

45 
54 
63 

89 
5.870, 

^10 

095 

248 

523 

72 

30 

4U 

45 
50 
55 

077 
059 
041 

195 
Mi 
089 

676 
828 

980 

8r 
89 
i      98 

44 
58 
69 

266 


GEODETIC  OPERATIONS. 


Lat. 

•Log  A. 

Log  B. 

LogC. 

Log  D.      Log  E. 

28°00' 

8.5094025 

8.511  7036 

1.13132 

2.3107   i   5-8786 

05 

006 

8.5116983 

284 

15        99 

IO 

.8.5093989 

930 

435 

24 

•  5-8813 

15 

970 

876 

586 

32        27 

20 

952 

823 

737 

4i 

4i 

25 

936 

768 

887 

49        56 

30 

918 

715 

i  .  14037 

57  i      70 

35 

899 

66  1 

187 

65  i      84 

40 

881 

608 

336 

74 

98 

45 

863 

552 

485 

82 

5-8912 

50 

845 

498 

634 

90        26 

55 

827 

444 

783 

98        40 

29  oo 

808 

390 

932 

2.3206        55 

05 

790 

335 

1.15080 

14        69 

10 

772 

281 

227 

22           83 

15 

753 

226 

375 

29           98 

20 

735 

171 

522 

37     5-9012 

25 

716 

116 

669 

45  •      26 

30 

698 

061 

816 

53  i      4i 

35 

679 

007 

963 

60        55 

40 

66  1 

8.5"5950 

1.16109 

68 

70 

45 

644 

896 

255 

75  i      84 

50 

624 

841 

401 

83        98 

55 

605 

737 

546        9°     5.9"3 

30  oo 

588 

728 

691 

98 

27 

05 

570 

672 

835 

2.3305 

42 

IO 

552 

616 

981 

12        57 

15 

533 

561 

1.17126 

19 

7i 

20 

5U 

505 

270 

27 

86 

25 

494 

449 

414 

34     5-9201 

30 

476 

394 

558 

4i        15 

35 

458 

337 

701 

48        30 

40 

439 

280 

845 

55        45 

45 

420 

225 

988 

62        60 

50 

401 

168 

1.18131 

69        74 

55 

376 

112 

274 

75  j      89 

31  oo 

361 

054 

416 

83     5-9304 

05 

339 

8.5114998 

578 

89        19 

IO 

324 

942 

700 

96        34 

15 

3°5 

884 

842 

2.3402  S      49 

20 

286 

826 

984 

09 

64 

25 

267 

769 

1.19125 

16 

78 

30 

248 

712 

266 

22 

93 

35 

229 

655 

407 

29 

5.9408 

40 

211 

598 

548 

35 

23 

45 

192 

539 

688 

4i 

39 

50 

173 

483 

829 

48 

54 

55 

153 

424 

969        54 

69 

FORMULA  AND  FACTORS. 


26; 


Lat. 

Log  A. 

Log  B. 

LogC. 

LogZ). 

Log*. 

32°oo' 

8.5093134 

8.5114367 

1  .  20109 

2.3460 

5-9484 

05 

"5 

309 

248 

66 

99 

10        096 

251 

388 

73 

5-95I4 

15      077 

193 

527 

79 

29 

20            057 

135 

666 

85 

44 

25             037 

077 

805 

60 

30        oi  8 

02O 

944 

97 

75 

35    8.5092998 

8.5113963 

1.21082 

2  3503 

90 

40        979 

903 

221 

09 

5.9606 

45         960 

844 

359 

14 

21 

50        940 

786 

497 

20 

36 

55         921. 

727 

635 

26 

51 

33  oo 

901 

669 

772 

32 

67 

05 

881 

611 

910 

37 

82 

10 

862 

552 

1.22047 

43 

98 

15 

842 

492 

184 

48 

5-9713 

20 

823 

434 

321 

54 

29 

25 

803 

374 

458 

59 

44 

30 

783 

315 

594 

65 

60 

35 

764 

257 

730 

70 

75 

40 

744 

197 

867 

76 

91 

45 

724 

137 

1.23003 

81 

5.9807 

5° 

704 

078 

139 

86 

22 

55 

684 

018 

274 

91 

38 

34  oo 
05 

665 
645 

8.5112959 
898 

409 
545 

97 

2  .  3602 

54 
69 

10 

625 

839 

680 

07 

85 

15 

605 

779 

8i5 

12 

5.9901 

20 

585 

720 

950 

17 

17 

25 

565 

660 

1.24085 

22 

32 

3° 

545 

600 

220 

27 

48 

35 
40 

525 
505 

540 
481 

353 
489 

32 

37 

64 

So 

45 
5° 

485 
465 

420 
363 

623 

757 

46 

96 

6.0012 

55 

445 

299 

891 

51 

27 

35  oo 
05 

424 
404 

238 
178 

1-25023 
157 

56 
60 

44 
60 

10 

383 

118 

290 

65 

76 

15 

20 
25 
30 

35 

363 
344 
320 
303 
283 

058 
8.5111997 
936 
875 
814 

424 

557 
690 
823 
955 

69 
74 

78 
83 
87 

92 
6.0108 
23 
40 
56 

40 
45 
50 

55 

263 
24-3 
223 
203 

753 
693 
633 
571 

i  .  26088 
220 
353 
485 

92 
96 
2.3700 
04 

72 
88 
6  .  0204 
21 

268 


GEODETIC  OPERATIONS. 


Lat. 

Log  A. 

Log£. 

Log  C. 

LogD. 

LogJe. 

36°oo' 

8.5092182 

8:5111509 

1.26617 

2.3709 

6.0237 

05 

161 

448 

749 

13 

53 

10 

141 

387 

881 

17 

69 

15 

121 

326 

1.27013 

21 

86 

20 

100 

265 

145 

25 

6.0302 

.25 

080 

203 

276 

29 

18 

30 

060 

142 

407 

33 

35 

35 

039 

080 

539 

37 

5i 

40 

018 

018 

670 

4i 

67 

45 

8.5091998 

8.5110957 

801 

44 

84 

50 

978 

895 

93i 

48 

6.0400 

55 

956 

834 

I  .  28062 

52 

17 

37  oo 

936 

772 

193 

56 

33 

05 

915 

710 

323 

60 

50 

10 

894 

648 

454 

63 

66 

15 

874 

587 

584 

66 

83 

20 

854 

525 

714 

70 

6.0500 

25 

833 

462 

845 

74 

16 

30 

812 

401 

975 

77 

33 

35 

791 

339 

1.29104 

81 

50 

40 

771 

276 

234 

84 

66 

45 

750 

215 

364 

87 

83 

50 

729 

151 

494 

9i 

6  .  0600 

55 

708 

090 

623 

94 

17 

38  oo 

687 

027 

753 

97 

33 

05 

667 

8.5109964 

882 

2.3800 

50 

10 

646 

902 

1.30011 

03 

67 

15 

625 

840 

140 

07 

84 

20 

604 

777 

09 

6.0701 

25 

533 

7*5 

398 

13 

18 

30 

562 

652 

527 

16 

35 

35 

541 

59° 

656 

18 

51 

40 

521 

526 

785 

22 

68 

45 

499 

463 

913 

24 

85 

50 

479 

401 

1.31042 

27 

6.0802 

55 

45? 

338 

170 

30 

19 

39  oo 

437 

275 

299 

33 

37 

05 

416 

212 

427 

35 

54 

10 

395 

150 

555 

38 

7i 

15 

374 

099 

683 

4i 

88 

20 

353 

023 

811 

43 

6.0905 

25 

S32 

8.5108960 

939 

46 

22 

30 

3n 

897 

1.32067 

48 

40 

35 

290 

843 

195 

51 

57 

40 

269 

770 

323 

53 

74 

45 

248 

707 

450 

56 

9i 

5° 

227 

644 

578 

58 

6.1009 

55 

206 

581 

706 

61 

26 

FORMULAE  AND  FACTORS. 


269 


Lat. 

Log  A. 

LogS. 

LogC. 

LogD. 

LogE. 

4o°oo' 

8.5091184 

8.5108518 

1.32833 

2.3863 

6.1043 

05 

I63 

455 

960 

65 

61 

10 

142 

393 

1.33088 

67 

78 

15 

125 

327 

215 

69 

96 

20 

099 

264 

342 

72 

6.1113 

25 

079 

2OI 

470 

74 

3° 

3° 

057 

137 

596 

76 

48 

35 

036 

073 

723 

78 

65 

40 

015 

OIO 

850 

80 

83 

45 

8.5090998 

8.5107946 

977 

82 

6.I2OI 

50 

972 

883 

1.34104 

84 

18 

55 

952 

820 

231 

86 

36 

41  oo 

930 

755 

358 

88 

54 

05 

909 

69I 

485 

90 

71 

10 

888 

628 

611 

gi 

89 

15 

867 

574 

738 

93 

6.1307 

20 

845 

500 

864 

95 

24 

25 

824 

437 

991 

96 

42 

30 

803 

373 

1.  35"7 

98 

60 

35 

781 

308 

244 

2.3900 

78 

40 

760 

244 

370 

OI 

96 

45 

739 

181 

497 

03 

6.1413 

50 

7i8 

117 

623 

04 

31 

55 

696 

053 

749 

06 

49 

42  oo 
05 

675 
653 

8.5106989 
925 

874 
1.36001 

07 
08 

67 

85 

10 

632 

861 

127 

IO 

6.1503 

15 

610 

797 

253 

II 

21 

20 

590 

733 

379 

12 

39 

25 

568 

668 

5°5 

14 

57 

30 

547 

604 

631 

15 

75 

35 
40 

524 
504 

541 

476 

757 
883 

16 

17 

94 
6.1612 

45 

483 

413 

1.37009 

18 

3° 

5° 

460 

348 

135 

19 

48 

55 
43  oo 
05 

439 
419 
396 

284 

220 
156 

261 
386 

5" 

20 
21 
22 

66 
85 
6.1703 

10 

376 

092 

638 

23 

21 

15 

354 

028 

764 

24 

4° 

20 

333 

8.5105963 

889 

25 

58 

25 

312 

899 

1.38015 

25 

76 

30 

35 

290 

269 

835 
771 

141 
266 

26 
27 

95 
6.1813 

40 
45 

247 
226  . 

706 
642 

392 

27 
28 

32 
50 

50 

55 

204 

183 

578 
513 

643 
769 

29 
29 

69 

87 

2/0 


GEODETIC  OPERATIONS. 


Lat. 

Log  .4. 

Log,?. 

LogC. 

LogZ>. 

Log£. 

44°00' 

8.5090162 

8.5105449 

1.38894     2.3930 

6.1906 

05 

140 

375 

1  .  3902.0         30 

24 

IO 

118 

3" 

145         31 

43 

15 

097 

256 

271 

31 

62 

20 

076 

193 

396 

32 

80 

25 

054 

128 

522 

32 

99 

30 

033 

063 

647 

32 

6.2017 

35 

on 

8.5104999 

773 

32 

36 

40 

8.5089990 

935 

998 

33 

55 

45 

969 

870 

1.40024 

33 

74 

50 

947 

806 

149 

33 

93 

55 

925 

741 

275 

33 

6.2II2 

45  oo 

904 

677 

400 

33 

3i 

05 

883 

612 

526 

33 

50 

IO 

861 

548 

651 

33 

69 

15 

840 

484 

777 

33 

88 

20 

8x8 

419 

002 

33 

6.2207 

25 

797 

356 

I.4IO28 

33 

26 

30 

776 

291 

153 

33 

45 

35 

754 

226 

279 

33 

64 

40 

733 

162 

404 

33 

83 

45 

711 

098 

530 

32 

6  .  2302 

50 

690 

034 

655 

32 

21 

55 

668 

8.5103969 

78l 

32 

40 

46  oo 

647 

905 

006 

3i 

60 

05 

625 

841 

1.42032 

3i 

79 

IO 

604 

776 

157 

30 

98 

15 

583 

712 

283 

30 

6.2417 

20 

56i 

648 

409 

29 

37 

25 

539 

584 

534 

29 

56 

3° 

518 

5i8 

660 

28 

76 

35 

497 

457 

786 

28 

95 

40 

475 

392 

911 

27 

6.2514 

45 

454 

326 

1.43037 

26 

34 

5° 

431 

262 

163 

26 

53 

55 

410 

199 

289 

25 

73 

47  oo 

390 

134 

414 

24 

93 

05 

368 

070 

539 

23 

6.2612 

IO 

347 

005 

666 

22 

32 

15 

326 

8.5102941 

792 

21 

52 

20 

304 

876 

917 

21 

71 

25 

283 

813 

1.44043 

20 

9i 

30 

261 

749 

169 

19 

6.2711 

35 

240 

685 

295 

17 

30 

40 

219 

621 

421 

16 

50 

45 

197 

557 

547 

15 

70 

50 

176 

493 

673 

14 

90 

55 

155 

428 

799 

13 

6.2810 

FORMULA.  AND  FACTORS. 


271 


Lat. 

Log  A. 

Log  B. 

LogC. 

Log/). 

Log.E. 

48°oo' 

8.5089133 

8.5102364 

1.44926 

2.39" 

6.2830 

05 

112 

301 

1.45052 

IO 

50 

10 

Ogi 

236 

I78 

09 

70 

15 

O7O 

172 

304 

08 

90 

20 

048 

108 

431 

06 

6.2910 

25 

027 

045 

557 

05 

30 

30 

005 

8.5101981 

683 

03 

50 

35 

8.5088984 

917 

809 

02 

70 

40 

963 

853 

937 

OO 

91 

45 

941 

789 

1.46063 

2.3899 

6.3011 

50 

92O 

725 

189 

97 

31 

55 

899 

662 

316 

95 

51 

49  oo 

878 

598 

442 

94 

72 

2/2 


GEODETIC  OPERATIONS. 


FROM    UNITED   STATES   COAST   SURVEY   REPORT. 

AUXILIARY     TABLES     FOR     CONVERTING  ARCS   OF    THE    CLARKE   ELLIPSOID   INTO 

ARCS   OF   THE    BESSEL    ELLIPSOID. 

[All  corrections  are  positive.] 


Corrections  to  dM. 

Arguments  L'  and  dM. 

dM 

60' 

50' 

40' 

3°' 

20' 

10' 

60" 

So" 

40" 

30" 

10 

5  ' 

Lat. 

23 

0.481 

0.401 

0.320 

0.240 

0.160 

0.080 

0.008 

0.006 

0.005 

0.004 

0.003 

0.00 

o  0006 

24 

.484 

•4°3 

.322 

.242 

.161 

.080 

.008 

.006 

.005 

.004 

.003 

.OO 

.0006 

25 

.486 

.405 

•324 

•243 

.162 

.081 

.008 

.006 

.005 

.004 

.003 

.00 

.0006 

26 
27 

.489 
.491 

.407 
.409 

-326 
•327 

IS! 

.163 
.164 

.081 
.082 

.008 
.008 

.006 
.006 

.005 
.005 

.004 
.004 

.003 
.003 

00 

.0006 
.0006 

28 

•494 

.411 

•329 

.247 

•  165 

.082 

.008 

.007 

.005 

.004 

.003 

.00 

.0006 

29 

.496 

.413 

•33° 

.248 

.166 

.083 

.008 

.007 

.004 

.003 

.00 

.0006 

3° 

•497 

.416 

•332 

.250 

.167 

.083 

.008 

•°°s 

.004 

.003 

.00 

.0006 

31 

.502 

.418 

•  334 

•251 

.168 

.084 

.008 

.007 

.006 

.004 

.003 

.00 

.0006 

32 

•505 

.420 

•336 

•253 

.169 

OS^ 

.008 

.007 

.006 

.004 

.003 

.00 

.0006 

33 

•5°7 

.422 

•338 

•254 

.169 

.085 

.008 

.007 

.006 

.004 

.001 

.0006 

34 

.51° 

•425 

•34° 

•255 

.170 

.085 

.008 

.007 

.006 

.004 

.003 

.OOI 

.0006 

P 

3 

.427 
•  43° 

•342 
•342 

-256 
.258 

.171 
.172 

.086 
.086 

.008 
.009 

.007 
.007 

.006 
.006 

.004 
.004 

.003 
.003 

.00 

.0006 
.0006 

37 

.518 

•432 

•345 

•259 

•'73 

.087 

.009 

.007 

.006 

.004 

.003 

.OOI 

.0007 

38 
39 

.521 
•524 

•  434 
•436 

•347 
•349 

:26i 
.262 

.174 
•'75 

.087 
.088 

.009 
.009 

.007 
.007 

.006 
.006 

.004 
.004 

.003 
.003 

.00 

.0007 
.0007 

4° 

•527 

•439 

•  35' 

.264 

.176 

.088 

.009 

.007 

.006 

.004 

.003 

.00 

.0007 

•530 

•353 

.265 

.177 

.089 

.009 

.007 

.006 

.004 

.003 

.00 

.0007 

42 
43. 

is 

•444 
.446 

•  355 
•357 

.267 
.268 

.178 
.179 

.089 
.090 

.009 
.009 

.007 

.006 
.006 

.004 
.004 

.003 

.00 

.0007 

.0007 

44 

•539 

•449 

•359 

.270 

.180 

.090 

.009 

.008 

.006 

.005 

.003 

.001 

.0007 

45 

0.542 

o  45' 

0.^61 

0.271 

o,  18 

0.091 

o.ooq 

0.008 

0.006 

0.005 

0.003 

O.OOI 

o  0007 

Corrections  to  dL. 

Arguments  —   —  and  dL. 

dL. 

60' 

50' 

40' 

3o' 

20' 

10' 

60" 

50" 

40" 

30" 

20" 

10" 

5" 

Lat. 

23 

0.193 

0.160 

0.129 

0.096 

0064 

0.032 

0.003 

0.00 

0.002 

0.002 

O.OOI 

O.OOI 

0.0003 

24 

.200 

.165 

•133 

.099 

.066 

•033 

.003 

.00; 

.002 

.002 

.001 

.001 

.0003 

11 

.206 
.213 

.171 
.177 

.138 
.142 

:a 

.068   .034 
.070   .035 

.003 

oo- 

!o02 

.002 

'ooi 

.001 
.001 

.0003 

27 

.220 

.183 

.147 

.110 

•073   -037 

.004 

.00; 

.002 

.002 

.001 

.001 

.0003 

28 

•     27 

.189 

.151 

•tt3 

.075   .038 

.004 

.00; 

.002 

.002 

.001 

.001 

.0003 

29 

•   34 

.196 

.156 

.117 

.o78|     .039 

.004 

.00' 

.OO2 

.OO2 

.OOI 

.OOI 

.0003 

30 

•   42 

.202 

.161 

.080 

.040 

.004 

.00; 

.002 

.002 

.001 

.001 

.0003 

31 

•  5° 

.209 

.167 

•  125 

.083 

.042 

.004 

.00; 

.003 

.002 

.001 

.001 

.0004 

32 

•   58 

.2l6 

.172 

.129 

.086 

.043 

.004 

.OO' 

.OO2 

.OOI 

.001 

.0004 

33 
34 

•275 

.223 
•230 

.178 
.184 

•  133 
•  137 

.089 
.091 

31 

.005 
.005 

.003 

.00; 

.002 

.002 

.001 
.00 

.0004 
.0004 

35 

.283 

.237 

.190 

.141 

.094 

.047 

.005 

.OO 

.002 

.002 

.001 

.0004 

36 

.291 

.243 

.195 

•  145 

.097 

.o48 

.004 

.OO 

.OO2 

.002 

.001 

.0004 

3l 

.300 

.250 

.150 

.050 

.005 

.004 

.OO' 

.OO2 

.002 

.00 

.0004 

38 

.308 

•257 

.206 

•  154 

.103 

.051 

.005 

.004 

.003 

.002 

.002 

.001 

.0004 

39 

•3*7 

.264 

.212 

.158 

.106 

.053 

.005 

.004 

.004 

.OO 

.002 

.00 

.0004 

40 

•325 

271 

.217 

.162 

.108 

.054 

.005 

.004 

.004 

.OO; 

.OO2 

.00 

.0005 

41 

•334 

.278 

.167 

.   ii 

.056 

.006 

.004 

.004 

.OO' 

.002 

.00 

.0005 

42 

•343 

.286 

.229 

•   i4 

.057 

.006 

.004 

.004 

.00' 

.002 

.00 

.0005 

43 

•352 

.294 

.236 

.176 

•  i? 

.006 

.004 

.OO' 

.OO2 

.00 

.0005 

44 

•  362 

.302 

.242 

.181 

.  20      .060 

.006 

.005 

.004 

.003 

.OO2 

.00 

.0005 

45 

0.372 

0-310 

0.249 

0.186    o.  24    0.062 

0.006 

0.005 

0.004 

0.003 

0.002 

O.OOI 

0.0005 

FORMULA  AND  FACTORS. 


273 


TAKEN    FROM    U.    S.    COAST   AND   GEODETIC   SURVEY    REPORT. 

SUBSIDIARY   TABLE    FOR    REFERRING   VALUES    OF    COEFFICIENTS   A,    B,   C,    D,    E, 

FROM  CLARKE'S  TO  BESSEL'S  ELLIPSOID. 


Lat. 

To  log  A  add. 

To  log  B  add. 

To  log  C  add. 

From  log  D 
subtract. 

To  log  E  add. 

23° 

0.0000582 

0.0000233 

0.00008 

0.0061 

O.OOOI 

24 

584 

N  241 

08 

6l 

25 

587 

249 

08 

6l 

26 

590 

258 

08 

61 

27 

593 

266 

09 

61 

28 

596 

274 

09 

61 

29 

599 

283 

09 

61 

30 

602 

293 

09 

61 

31 

605 

302 

09 

61 

32 

609 

312 

09 

61 

33 

612 

321 

09 

61 

34 

615 

331 

09 

61 

35 

619 

342 

IO 

61 

36 

622 

352 

10 

61 

37 

625 

362 

IO 

61 

38 

629 

372 

IO 

61 

39 

632 

383 

IO 

61 

40 

636 

393 

10 

61 

4i 

639 

404 

IO 

61 

42 

643 

II 

61 

43 

647 

425 

II 

61 

44 

650 

436 

II 

61 

654 

447 

II 

61 

46 

657 

458 

II 

61 

47 

661 

468  . 

II 

61 

48 

664 

479 

II 

61 

49 

668 

490 

12 

61 

50 

672 

501 

12 

61 

TABLE   OF   LOG  F. 


Lat. 

Log  F. 

Lat. 

Log-F. 

Lat. 

Log  .P. 

Lat. 

Log^1. 

23° 

7.812 

30° 

7.866 

37° 

7.876 

44° 

7.848 

24 

23 

31 

70 

38 

74 

45 

40 

25 

32 

32 

73 

39 

72 

46 

32 

26 

41 

33 

75 

40 

69 

47 

24 

27 

49 

34 

77 

4i 

64 

48 

*4 

28 
29 

55 
61 

11 

77 
77 

42 
43 

60 
54 

49 
50 

04 
7.792 

IS 


274 


GEODETIC  OPERATIONS. 


TABLE   OF  CORRECTIONS   TO   LONGITUDE   FOR   DIFFERENCE 
IN   ARC   AND   SINE. 


Log  K(-\ 

Log  difference. 

Log  dM(+\ 

Log**-). 

Log  difference. 

Log  dM(+). 

3.871 

0.0000001 

.330 

4-913 

O.OOOOirg 

3.422 

3-970 

OO2 

-479 

4.922 

124 

3-431 

4-"5 

003 

.624 

4-932 

130 

3-441 

4.171 

004 

.680 

4.941 

136 

3-450 

4.221 

005 

•  730 

4.950 

142 

3-459 

4.268 

006 

•777 

4-959 

147 

3-468 

4.292 

007 

.801 

4.968 

153 

3-477 

4.309 

008 

.818 

4.976 

1  60 

3.485 

4.320 

009 

•839 

4-985 

1  66 

3-494 

4.361 

OIO 

.870 

4-993 

172 

3.502 

4-383 

on 

.892 

5.002 

i?9 

3-5II 

4-415 

012 

.924 

5.010 

1  86 

3-519 

4-430 

013 

•939 

5-017 

192 

3-526 

4-445 

014 

•954 

5-025 

199 

3-534 

4-459 

015 

.968 

5-033 

206 

3-542 

4-473 

016 

.982 

5  040 

213 

3-549 

4.487 

017 

.996 

5-047 

221 

3-556 

4-500 

018 

3-009 

5-054 

228 

3-563 

4-524 

020 

3-033 

5.062 

236 

3-571 

4.548 

023 

3-057 

5.068 

243 

3-577 

4-570 

025 

3-079 

5-075 

251 

3-584 

•591 

027 

3.100 

5.082 

259 

3-591 

t  .612 

030 

3-I2I 

5.088 

267 

3-597 

.631 

033 

3-140 

5-095 

275 

3-604 

.649 

036 

3-158 

5.102 

284 

3.611 

.667 

039 

3.176 

5.108 

292 

3-617 

4.684 

042 

3-193 

5-"4 

300 

3-623 

4.701 

045 

3.210 

5.120 

309 

3  629 

4.716 

048 

3-225 

5.126 

318 

3-635 

4-732 

052 

3-241 

5-132 

327 

3-641 

4.746 

056 

3-255 

5.138 

336 

3.647 

4.761 

059 

3-270 

5-144 

345 

3.653 

4-774 

063 

3-283 

5-150 

354 

3-659 

4.788 

067 

3  297 

5-156 

364 

3-665 

4.801 

071 

3-3io 

5.161 

373 

3.670 

4.813 

075 

3-322 

5-167 

383 

3.676 

4-825 

080 

3-334 

5-172 

392 

3.681 

4-834 

084 

3-343 

5.178 

402 

3.687 

4.849 

089 

3-358 

5-183 

412 

3-692 

4.860 

094 

3-369 

5.188 

422 

3-697 

4.871 

098 

3-380 

5-193 

433 

3.702 

4.882 

103 

3-391 

5-199 

443 

3.708 

4.892 

108 

3.401 

5-204 

453 

3-7I3 

4-903 

114 

3-412 

INDEX. 


ABULFEDA,  description  of  Arabian  arc-measurement, 3 

Adjusting  the  azimuth 200 

Adjustment,  figure 168 

station 146 

when  directions  have  been  observed 180 

Airy 248 

Alexandria 2 

Anaximander i 

Angles,  method  of  measuring 97 

Arabian  arc-determination 2 

unit  of  measure 28 

Arago 15 

Argelander 18 

Auzout 29 

Axes  of  the  earth,  Clarke's  and  Bessel's  values  of 259 

Azimuth,  affected  by  adjustment 200 

formula  for  computing 218 

BAEYER 20 

Barrow,  Indian  arc-measurement 13 

Base  apparatus,  first  form  of 50 

requisites  of 50 

Bache-Wurdeman 52 

Baumann. 60 

Bessel 58 

Borda 51 

Colby 59.  64 

Ibafiez 60 

Lapland •  • So 

Peru 50 

Porro 59 

Repsold 60 

Struve 59 


2/6  INDEX. 

PAGE 

Base-line,  probable  error  of 74 

reduction  to  sea-level 78 

Base-measurements 49 

aligning 64 

comparison  of  results 61 

computation  of  results 70 

erection  of  terminal  marks 66 

general  precautions 69 

inclination 68 

instructions 64 

sector  error 68 

selecting  site 64 

record,  form  of •    66 

references 79 

transference  of  end  to  the  ground 67 

Beccaria,  arc  measurement to 

Bessel 247 

base  apparatus 20 

review  of  the  French  arc 15 

Biot 15 

Bonne 19 

Borda,  metallic  thermometer 14 

Borden,  survey  of  Massachusetts 23 

Boscovich,  arc-measurement 10 

Bouguer 7 

Boutelle 81,  86 

Brah6,  Tycho 29 

Briggs 29 

CALDEWOOD,  glass  base  apparatus n 

Camus 8 

Cape  of  Good  Hope  arc 10 

Cassini 4,  II 

revision  of  the  French  arc 9 

Celsius 8 

Centre,  reduction  to 196 

Chaldean  unit  of  measure 28 

Chauvenet 160 

Circle,  entire,  first  used 29 

Clairault 8 

theory  of  the  figure  of  the  earth 9 


INDEX. 


Clarke,  solution  for  the  figure  of  the  earth  ........................ 

reference  to  the  great  English  theodolite  ....... 

Coast  Survey,  U.  S.,  organized  ................................  '""     '     2J 

form  of  base  apparatus  .........................  52,  61 

heliotrope  .................................  '  46 

signals  ....................................  g4 

theodolite  .............................  ____  32 

Colonna  ..................................... 

Commission  for  European  degree-measurements  ........................  26 

Comparison  of  base-bars  with  a  standard  ..............................  j! 

Condamine,  De  la  ...........................................  _ 

Connection  of  France  and  England  by  triangulation  .....................  n 

Constants  and  their  logarithms,  table  of  ..............................  260 

Correction  for  inclination.    ............................................  -4 

Correlatives,  equations  of  .............................................  jjg 

Cutts  ................................................................  86 

DAVIDSON  ................  .  .......................................  47,  102 

Delambre  ............................................................     !3 

revision  of  the  Peruvian  arc  ..................................       8 

Des  Hays,  pendulum-investigations  ............................  ........       6 

Directions,  adjustment  of  ..........  .  .................................   180 

horizontal,  copy  of  record  ..................................   101 

Dividing-engine  first  used  ...........................  ..................     30 

Dixon.     See  Mason. 

Doolittle  ................................  ,  ............................  195 

ECCENTRIC  signal  ....................................................  146 

instrument  ..................................................  196 

Ellipticity  of  the  earth,  table  of  ........................................  249 

Errors,  mean  of  ......................................................  146 

probable.     See  Probable  Errors. 

Equations,  conditional,  number  of  .....................................  179 

side  ......................................................  171 

solution  of,  by  logarithms  ...................................  194 

Eratosthenes  .........................................................  I 

Everest,  Indian  arc  ...................................................  14 

Expansion  coefficient,  determination  of  .................................  72 

FERNEL  .............................................................  3 

Figure-adjustment  ....................................................  168 

Figure  of  the  earth  ....................................................  234 

literature  of  .......  ................................  249 


2/8  INDEX. 

PACK 

French  Academy  arc-measurements  in  Lapland  and  Peru 6 

Froriep I 

GASCOIGNE,  first  to  use  spider-lines 4 

Gauss 19,  159 

Geodesic,  Ecole  speciale  de 17 

Geodetic  factors,  tables  of 264 

Godin 7 

Greek  unit  of  measure 28 

HANSTEEN 18 

Heights  determined  by  barometer 94 

triangulation 101 

Heliotrope,  description  of 45 

first  used 45 

illustration 46 

use  and  adjustments 46 

Hilgard 41 

Hipparchus 29 

Hounslovv  Heath  base  .    12 

Humboldt 29 

Huygens's  theory  of  centrifugal  motion 6 

INGENIEUR  Corps,  organization  of 16 

Instruments 28 

Invention  of  the  vernier 29 

Isle,  De  1' 17 

Italian  commission  organized 25 

Italy,  co  operation  with  Switzerland 17 

James,  Sir  Henry,  reference  to  the  great  English  theodolite II 

LACAILLE,  revision  of  Picard's  arc 4 

arc-measurement  at  the  Cape  of  Good  Hope 10 

Lahire 4 

Lambton,  Indian  arc 14 

Lapiace 13,  247 

Lapland  arc-measurement 8 

Latitude,  formula  for  computing 203 

illustration 220,  222 

Least  squares,  theory  of 104 


INDEX.  279 

Letronne,  review  of  Posidonius's  arc 

Level,  table  giving  difference  between  the  true  and  apparent 92 

Liesganig,  arc-measurement lo 

Littrow 

Longitude  determined  by  powder-flashes IQ 

Longitude,  formula  for  computing 217 

illustrations 22O)  22~ 

MACLEAR,  continuance  of  the  Cape  of  Good  Hope  arc 23 

Maraldi "" 

Mason,  Maryland-Pennsylvania  boundary-line lo 

Maupertuis,  Lapland  arc 3 

Mayer,  repeating-theodolites I2 

Mechain n,  15 

Metre,  determination  of  its  length !3 

legal  and  recent  values  of 249 

Metres  to  feet,  table  for  converting 257 

miles 258 

yards 257 

Micrometer  first  used 29 

determination  of  run 35 

Miles  to  metres,  table  for  converting 259 

Monnier 8 

Mosman 97 

Mudge 15 

Muffling,  Von , 19 

Musschenbroeck 19 

NAPIER : 29 

Newton,  theory  of  universal  gravitation  demonstrated  by  Picard's  arc 4 

Normal  equations 155 

Normals,  table  of 261 

Norwood,  measurement  of  the  distance  from  London  to  York 4 

Nunez 29 


OUTHIER 


PALANDER 14 

Pamlico-Chesapeake  arc 24 

Phase,  correction  for *44 


28O  INDEX. 

PAGB 

Picard's  triangulation 4 

Pcle,  selection  of,  in  figure-adjustment 179 

Posch,  view  of  Ptolemy's  length  of  a  degree   2 

Posidonius,  arc-measurement 2 

Probable  error  of  the  arithmetical  mean 124 

of  a  single  determination 120 

illustration 122 

of  a  base-line 125 

direction 125 

in  the  computation  of  unknown  quantities  in  triangles  127,  133, 

136,  137,  138 

Prussia,  first  geodetic  work 16 

Prussian-Russian  connection 25 

Ptolemy,  value  of  earth's  circumference 2 

Puissant,  review  of  the  French  arc 15 

Pythagoras I 

QUADRANT  of  the  earth,  length  of    249 

RADIUS  of  curvature,  table  of 261 

Reduction  to  centre 196 

Reichenbach 26,  31 

Repetition  of  angles,  principle  first  announced 12 

abandoned 31 

Repsold 26,  31 

Riccioli 3 

Richer,  expedition  to  Cayenne 6 

Roemer 29 

Roy ii,  15 

Russian  arc,  accuracy  of 18 

Russia,  first  geodetic  work 17 

SAEGMCLLER,  principal  of  bisection 31 

Schott 24,  179,  248 

Schmidt 247 

Schumacher 10,  yi 

Schwerd ig 

Series,  binomial  256 

exponential 256 

logarithmic 257 


INDEX  28l 

PAGB 

Signals,  form  used  on  coast-survey 84 

night  cost  of 82 

method  of  erection 87 

size  and  lengths  of  timbers 88 

Snellius'  triangulation 3 

Spain,  first  geodetic  work 25 

Spherical  excess,  computation  of 168 

Speyer  base 10, 

Spider-lines  in  telescope,  first  use  of 4 

Stations,  description  of 05 

permanent  markings 96 

intervisibility  of go 

Station-adjustment ...  146 

Struve 17 

S  van  berg 14 

Sweden,  coast  triangulation    26 

Swedish  arc-measurement  in  Lapland 14 

Switzerland,  first  geodetic  work 17 

Syene i 

TENNER 17 

Thales i 

Theodolites,  adjustment  of 33 

construction  of 32 

errors  of  eccentricity 36 

graduation 39 

illustration 34 

size  of 32 

Toise  of  Peru 7 

Transferring  underground  mark  to  the  top  of  a  signal 96 

Triangles,  best  composition  of 86 

Triangulation,  calculation  of 143 

conditions  to  be  fulfilled 143 

Trigonometric  expressions 253 

series 256 

ULLOA 7 

VARIN,  pendulum-investigations 6 

Vernerius 29 


282  INDEX. 

fAGK 

Vernier,  invention  of 29 

distance  apart 37 

WALBECK  247 

Walker 22 

Waugh 22 

Weights,  application  of,  in  adjustments 165 

YARDS  to  metres,  table  for  converting 258 

Yollond 184 

ZACH,  VON,  revision  of  Beccaria's  arc 10 

of  the  Peruvian  arc. . .  8 


